PROBLEMAS OBA PDF
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But in this case. They have a common side BC. C are two opposite vertices. Let us compare triangles Solution. A triangle which has two equal angle bisectors is isosceles.
Therefore CC 0 are equal. We have just have angle bisectors. Then BB 0 CF is a parallelogram. If ta and tb are the lengths of the angle bisectors of angles A and B of triangle ABC with sides of length a.
Here it is solved by considering inequalities. Now if two angles are unequal. Suppose BB 0 and CC 0 are these two equal angle bisectors. In any triangle. If we exchange the roles of points B and C. Problem b. This is a classically difficult problem. The more general problem will be solved if we can locate point D such that D1 D2 is as short as possible. AB of the triangle. We note first that. They can also explain why the statement in the problem implies that a triangle with two equal angle bisectors must be isosceles.
D2 figure ta. AC respectively. F lie respectively on sides BC. Figure ta Let us first fix point D on line BC. We construct points D1. Thus we can construct this triangle by reflecting D to get D1 and D2. Of all the triangles inscribed in a given triangle. The problem is solved. Show that the problem does not have a proper solution that is. And A is also on line AC. The same reasoning applies to vertices E and F: Thus the required triangle is indeed the one formed by the feet of the altitudes of the original triangle.
Because the angle at A in isosceles triangle AD1 D2 does not vary. We have already seen that one vertex of the triangle of minimal perimeter is a foot of an altitude of triangle ABC. In a quadrilateral ABCD. But if ABCD is cyclic. Show that for a given cyclic quadrilateral. Figure tbi Then we have: The perimeter of the minimal inscribed quadrilateral is the fourth proportional for the radius of the circle ABCD and the two diagonals AC.
For this case. Suppose quadrilateral M N P Q. We break the problem statement into several parts: If there exists a quadrilateral of minimal perimeter inscribed in a given quadrilateral.
For minimal inscribed quadrilateral M N P Q itself to be cyclic. C 0 D If this is the case. Point N reflects into N 0. We will use a proof based on geometric transformations. By Figure bii shows the original quadrilateral ABCD reflected four times: P 0 and Q0 are all collinear. Clearly these four new quadrilaterals are congruent to the original one. Let us look at the successive images of its vertices in the four reflections we consider above. Now we use the assumption that ABCD is cyclic.
But it is not difficult to see that there are still infinitely many positions of Q which yield a solution. Students can think about how the proof fails if ABCD is not to be cyclic. This completes the construction. We do this through a rather artificial construction.
Note that this solution can be seen as a generalization of the solution to exercise We see that there are infinitely many minimal quadrilaterals. We erect perpendiculars to. This computation is a difficult one.
For some positions of Q. C are collinear. Note that. DA are equal. We first show that A.
D are collinear. Then triangles BB 0 N. A0 lies on QA0. EDC are similar. Likewise we can show that A is on the bisector of SQM. Let S be the intersection of lines M N. These four hard-earned relationships involve pieces of the line segments we are interested in. SQ which are symmetric in this diagonal. Note that for any position of QQ0 that is. Choose a point S on one of the diagonals of the original quadrilateral. We base our solution on the result of exercise We apply this result to quadrilateral KN LO fig.
But from cyclic quadrilaterals BM B 0 N. Looking at triangle B 0 C 0 T. The intersection of these rays with the sides of the original quadrilateral are the vertices of a minimal inscribed quadrilateral. In the course of this discussion. L are the respective midpoints of M N.
The construction of M N P Q shows us that its sides always retain their directions. Applying the theorem of to quadrilateral AM BI. To evaluate this minimal sum. This the midpoints K. Show that the point obtained in Exercise K 0 of these segments lie along the median of any one of these triangles. Point K also slides along a fixed line.
N L retains its direction. Its square is half the sum of the squares of the three sides. If it is inside the triangle. What happens when the point is outside the triangle? Suppose M is any other point on the plane. In this case the point which minimizes the sum of the distances to the vertices of the triangle is the vertex of its obtuse angle.
L slides along another line. M 0 N 0 B are homothetic. We now show that N. CC 0 whose concurrence is proven in exercise And since KO. LO are perpendicular to these two segments. C 0 are the vertices of equilateral triangles constructed externally on the sides of a given triangle ABC.
We apply to both of these: We separate this problem into several statements: The result of exercise then shows that point O slides along a fixed line. L slide along fixed lines. Point N certainly slides along line BC. We draw the circumcircle fig. Using this last result. Proof due to Behzad Mehrdad. In case i. Adding these two results. C may form a quadrilateral in a different order.
For other positions. Figure tb shows only one possible position of point M. A0 BOC. If this point is not inside the triangle. This proves that lines AA0. All of these angles C. Lines AA0. CC 0 are concurrent at O. Let M be any point on the plane fig. Let this triangle be T. We assume first that the three given numbers can represent the sides of a triangle.
The construction described gives a single point the lines mentioned are concurrent. BA we construct angles CBA. If this point lies inside the triangle. A0 are collinear. In the first case. We will first show that point O lies on circle ABC 0 as well.
Figure ta shows triangle T and the construction described in the problem statement. BC 0 AO are all cyclic quadrilaterals. If it is not. CC 0 intersect at a point O. In the same way and using the fact that AB 0 CO. From cyclic quadrilateral A BCO. C 0 are collinear. A0 BC are similar. But by construction. We must express the length of AA0 in terms of the lengths a. This argument generalizes the result of exercise In both cases. This follows from pairs of similar triangles. Which one is actually the minimum?
Hence we have From the Pythagorean theorem. But if we follow the same logic using BB 0 or CC 0. We let A00 be the reflection in line BC of point A0.
An easy special case occurs when M and A0 coincide.
ACA 0 intercept the same arc on the circle through A. The argument here generalizes that of exercise We now give a computation of the minimal value. We now rewrite this last equation in terms of the sides of triangle T. We draw circle ABC. We will get close to the expression given in the problem statement if we express this last relationship in terms of the sides of the original triangle.
Suppose O lies outside triangle ABC. T are similar. Since we need to involve the area of ABC. Now adding 7 and 9. As is often the case when we use metric relationships in a triangle.
The theorem of assures us that there are two such points. IB 00 C 00 are similar. But quadrilateral ACIB is cyclic. Therefore in triangles T. BB 0 must lie outside the triangle.
Together with From the triangle inequality CF are concurrent. We divide each side of a triangle into segments proportional to the squares of the adjacent sides. Show that: Now by Let O0 be their point of intersection.
That this is precisely the point that would be obtained in Exercise The product of these three ratios is 1. This will tell us about the relationship between the two points. We first compute the ratio O0 P: O0 R where P. Using absolute value to denote area.
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Then quadrilateral BM CI may or may not be cyclic. The three lines obtained in this way are concurrent. The last condition is stronger than the first. AO0 C each pair of triangles has equal altitudes from B or C. That this point is the center of mass of the triangle P QR formed by its projections on the sides of the original triangle.
Hence OS: OCA are equal in area. CO in the corresponding angle bisectors of ABC. We can write this proportion as ABD: AO0 C. From these proportions. CO0 are symmetric to AO.
Now we make a similar. Hence ABO0: This implies that O0 is the point obtained from O by the construction of exercise It follows that O0 Q: OU are the distances from O to AB. But ABD: It now follows from the second lemma in the solution to exercise that lines AO0. A bit of algebra shows that this is equivalent to GJ: Now for fixed points K. The centroid of that triangle point O0 in exercise is the point such that the sum of the squares of its distances to the three sides is minimal.
In a given triangle. Conclude that the point O0 in the preceding exercise is the one such that the sum of the squares of its distances to the three sides is the smallest possible Exercises We first use to relate the sum of the squares of the sides to the lengths of the medians. If such a triangle exists. We will use arguments similar to those of exercise As in exercise We divide the problem into three parts.
Triangle P QR of exercise is the inscribed triangle such that the sum of the squares of its sides is minimal.
Since we can do this for any two vertices of JKL. Now let JKL be the inscribed triangle such that the sums of the squares of its sides is minimal so that its medians are perpendicular to the sides of ABC. This proves our lemma. Solution I: As suggested in the problem statement.
Assuming that this minimum exists. The result of 56 tells us that G0 J 0. G0 L0 are each 32 the corresponding medians. AB respectively.
L0 be the feet of the perpendiculars from M to BC. Suppose G0 is the centroid of triangle J 0 K 0 L0. This concludes the proof of the first assertion in the problem.
We now show that G is the point which minimizes the sum of the squares of the distances to the sides of ABC. Let M be any point in the plane fig. We express this relationship using the sides of triangle J 0 K 0 L0. This last relationship gives us everything we need. We choose point J1 such that K 0 J1: We can prove this as in the argument in the solution for exercise L0 L1 are concurrent at some point O0.
But that ratio is just J1 L0: L0 in place and moving J1 to coincide with the foot of the perpendicular from J 0 to line BC. If J1 J 0 were not perpendicular to BC. L 0 AK 0 are supplementary. That is: It follows that: We assume that the required triangle is J 0 K 0 L0 fig. It turns out useful to consider the following cases: We can then derive equation 5 above just as before.
Analogous derivations. We will make use of directed line segments. L0 on the sides of triangle ABC. L0 K1: Note 1.
This can be written as: This expression shows that if we move point H by moving point J 0 along line BC. For the cases in which the sums indicated in the previous note all have the same sign. But if we fix J 0. Our second solution. But even if we make the assumption that s cannot take arbitrarily small values as we vary triangle J 0 K 0 L0.
But in fact there is no such case. Then an examination of equation 6 shows that if we fix K 0. L2 be the feet of the perpendiculars from its centroid O0 to the sides of ABC. We can still imagine a case in which. Note 2. We start with the following algebraic identity. For our solution. Solution II. This equation holds for any six real numbers. Note that in the most general case. As in our first proof. We will show again that s can take on neither a maximum nor a minimum for any triangle J 0 K 0 L0.
As in the derivation of equation 3 above. This happens when O0 is the point we found in exercise The second sum is minimal when the pairs of points J2. Translating this to the geometric situation. As argued earlier. L0 coincide. We can write equation 7 in the form: We will use the following generalization of identity 7: O0 L2. L4 are the intersections of these perpendiculars with sides K3 L3.
We will show that the triangle we seek is the one formed by the feet of the perpendiculars P J3. As in the previous proofs. J3 K3 of J3 K3 L3 fig. Figure te Indeed. P L3 J3: Now it is not hard to see that P J4 K3: J4 L3.
L3 and examine similar triangles to show that the altitudes to common side P J4 of these triangles. L3 J3. P L3 to the sides of ABC. Thus P K3 L3: So far. LL0 are concurrent at some point O0. Also by direct computation. Hence K3 J4: To this end. L3 K4: This time we get: L0 in the ratios m: Z be the feet of the perpendiculars from O0 to BC.
Note that either side of 17 is in fact the value of s the quantity we are required to minimize. Rewriting 17 using these values.
In this sense. Equation 17 is very close to what we need. The argument that led to equation 12 actually contains a proof of a statement interesting in its own right.
If segments AL. We first relate LJ to BX. We draw BX. From similar triangles JLC. XBC we have JL: L will coincide with X. CN are concurrent at point J with points L. N on the sides of triangle ABC. Hence we see that if ABC is the triangle which maximizes s. Substituting these values into 18 gives us our result. We now relate these ratios to those in the problem statement. We also assume. Equation 17 gives the relationship between the bases of a trapezoid and half the line through its diagonals.
Solution I. The reasoning leading to equation 17 is similar to the argument in the solution of Exercise For this solution. From similar triangle AM J. In a given circle. Thus we have the following. If necessary. Now if we fix B and C.
This argument is analogous to the one in the last paragraph of the solution to exercise Here the two given circles in that exercise coincide. When does this construction work? It is necessary and sufficient that a point D. Either can be a position of B. We replace AD with its equivalent as given in equation 4. We choose a point A arbitrarily. Both of these are simple consequences of equation 1. We find point D which divides BC in the ratio m: As in the first solution.
Our second solution does not have this failing. We draw diameter M N through D. We now express s in terms of M D m and AD. We substitute this into 2 to obtain: See the note to that solution. This first solution has the same failing as the first solution we gave to exercise We can interpret this condition geometrically. But even this condition may not quite be sufficient: Substituting the value of MD given by 8. So we can ignore this part of the expression.
M is the most distant point on the circle from D. If point D. We can eliminate this second dependence. But this condition is not sufficient. We must also be sure that there exists a point D such that BD: It is not hard to see 64 that this condition is equivalent to DN: F 0 on the corresponding sides.
Suppose there exists a point D such that AD: C respectively in triangle ABC. These two continued proportions imply that AD: We have solved this problem in the discussions above. Show that the sum of these segments is constant. C onto points A0. OO1 D0 are clearly similar.
First we show that the indicated condition is necessary. F on the opposite sides of the triangle so that the segments AD. Through an interior point O of the triangle we draw segments OD0. Direct computation of these ratios will show that A0 B 0: By the result of exercise CF are equal. It is not hard to show that point D satisfies the requirements of the problem. We invert these three points in a circle around D.
C 0 which form a triangle congruent to T. The result of exercise b assures us that there is an inversion with some center D and some power of inversion whose value will not concern us which takes points A. If the image of points A. This problem is resolved by the discussion above. This exercise generalizes the result of exercise To show that the given condition is also sufficient.
We join the vertices of a triangle ABC to points D. Let the distances from O to BC. AB be x. These numbers are the sides of triangle A0 B 0 C 0.
C 0 respectively. A necessary and sufficient condition for the existence of a solution to Exercise a point whose distances to the three vertices of a triangle ABC are proportional to three given numbers m.
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C are A0. It follows that the coefficients AB. This proves the original assertion. AB in this expression for z do not depend on the position of M. Now if we move the position of M the shape of triangle OAB does not change: Figure t shows lines a. In this problem. The distances from M to a.
We first treat a simple case. Formulate the result in a manner entirely independent of the position of the point by an appropriate convention for the signs of the segments. We draw line c0 through M parallel to c. When three lines are concurrent. For some positions. So in general.
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A OA more general assertion. To formulate the result still more generally. For this. OB retains its magnitude and sign.
We will also need some further conventions about signs. The original assertion allows for these coefficients to be positive or negative for different positions of M. We want to say. We make analogous conventions for the distances y and z from M to lines b and c respectively. Using these conventions. For different positions of M in the plane. Suppose the two given lines intersect.
The preceding exercise provides a solution of the problem for n lines. From the converse proposition in exercise Find the locus of points such that the sum of their distances to n given lines. Deduce that the midpoints of the three diagonals of a complete quadrilateral are collinear. Students can move M about the plane and observe that triangle OAB retains its shape.
Let x1. This case reduces to the previous. If O is again the intersection of a and b. We use the conventions for directed line segments given in the solution to exercise OB along lines a.
They can also observe that sometimes areas must be added. We then draw a line c through O parallel to AB and let z be the signed distance from M to c. Let k1. The truth of the various assertions using conventions about signed areas and segments can thus be conveniently examined. Let a1. Now suppose the two given lines are parallel. Substituting in 1.
Dynamic geometry software can be useful in exploring this sort of problem. So in this case. If X is a point satisfying 1. Of any set of more than two non-zero numbers. As in the previous exercise. By our induction hypothesis. We can assume that no ki is zero. We now proceed by induction. The sign of x2 then reverses as well. As noted in the problem statement. Suppose we have line segments A1 B1.
A2 B2. An Bn. N be the midpoints of its diagonals fig. We add the first two of these equations. Equations 4.
Now we note that. N all belong to the locus of points X such that.
We prove the assertion by showing that point N belongs to this locus as well. This takes a bit more work than 4 and 5 did. ADF have the same powers with respect to these three circles. The simplicity of this argument belies the complex nature of the result. Triangle BCF is formed by three of the sides of the quadrilateral. This axis passes through the intersection of the altitudes of each of the four triangles formed by three sides of the quadrilateral.
Similarly L lies on the circle with diameter CD. In just the same way. For these points. If the circles intersect. Hence 7 in fact does determine a line. A subtle point completing this argument is to show that equation 7 does not hold for every point on the plane. Quadrilateral BCKL is cyclic two opposite angles are right angles.
So each of the four orthocenters lie on the intersection of the radical axes of pairs of these circles. The three circles whose diameters are the diagonals of a complete quadrilateral have the same radical axis. L lie on the same circle not shown and It is not hard to see that these conditions imply that quadrilateral ADBC is a parallelogram.
Since this locus is a line. If it did. We can read equation 1 as saying that H has the same power with respect to both these circles. These same lines intercept three segments on an arbitrary transversal such that the segment which divides two of them harmonically also divides the third harmonically. Let P QRS be four vertices of the given quadrilateral.
The opposite sides of a complete quadrilateral. The result is given in figure tb. The complete quadrilateral determined by P QRS is transformed into another complete quadrilateral.
We take the polar of each point in figure ta. Point K is transformed into line k. Transform by reciprocal polars. Three segments with this property are said to be in involution. Let opposite sides P Q. RS intersect at K. Thus points pq. OK form a harmonic pencil. But these three midpoints are collinear Figure tc To prove the second assertion of the problem. It follows from the first assertion of this problem that the polar of U with respect to the third pair P R.
QR is transformed into the midpoint of another diagonal of pqrs. This proves the first assertion of the problem. Since the harmonic conjugate of a point at infinity. QS of the original complete quadrilateral is transformed into the midpoint of the third diagonal of pqrs.
QS of lines also passes through V. The definition of the polar of a point with respect to an angle tells us that the polar of point U with respect to the pair of lines P Q. Therefore segment U V divides segment CC 0 harmonically. A third iteration of this argument will show that the polar of O with respect to the angle determined by diagonals P R. M is the midpoint of one of the diagonals of complete quadrilateral pqrs. Therefore the polars of these three lines. RS passes through point V.
Under the transformation by reciprocal polars. Suppose the segments these lines determine along T are AA0. There are three assertions here: Z be the feet of the perpendiculars from P to its three sides. Let H be the orthocenter of the triangle. Deduce from this.
BB 0 do not intersect. H 0 lies on the circumcircle of ABC. Figure ta To show this. The solution to problem b shows that points U and V exist if and only if the circles on diameters AA0. This is a condition on the position of line t.
We will first show that Q is on a line through H parallel to XY. The Simson line Exercise 72 which joins the feet of the perpendiculars from a point P on the circumscribed circle of a triangle to the three sides. We have point P and the feet K. We will prove. We fix points A. But we can tell more about P CY. P in line BC. Since P is on the circumcircle of triangle abc. We also have point P 0 and the feet K 0. P 0 move along S so that the distance P P 0 remains constant.
SaniZe, v. JRenti, l. Nnaiseri a. SaniZe, s. Ziris reduplikacia da sufiqsacia morfologiuri xerxebia, ZirSi an marcvalSi mom- xdari fonologiuri cvlilebebi ki reduplicirebis fonologiuri gzaa.
CVC 2 - tipi 2. TanxmovanTa sigrZis gazrdis SesaZlebloba exeba marcvlis struq- turasac. Cveni azriT, qarTulis araidenturxmovniani reduplikaciis am tipSi a an i xmovani imTaviTve maerTis funqciisa unda yofiliyo. Tandilava : a. Tandilava, lazuri leqsikoni, Tbilisi.
Tofuria : v. Metzler, Stuttgart-Weimar. JRenti s. Tbilisi qiria da sxv. Cuxua, lazur-megruli 60 gramatika, Tbilisi.
SaniZe : a. SaniZe, qarTuli enis gramatika I morfologia , Tbilisi. Stolci da sxv.Note that O1 stays fixed as P. The topic of our discussion table is Georgian IDP population, N6 15 '07 their social conditions, problems and developmental perspectives. L slides along another line.
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