## Monday, October 10, 2011

### Geometrical Problems

Finally, I've reached a point when I can start introducing problems in Geometry. Problems below are, mostly, from more than 100 years old textbook on geometry by a famous Russian math teacher Kiselev.
Solving problems is the most important purpose of our site UNIZOR.COM. Please examine these problems and try to solve them. The accompanying lecture contains all the solutions, but watch it only after you spent sufficient amount of time trying to solve these problems yourself.

1. Prove that in an isosceles triangle two medians to congruent sides are congruent.

2. Prove that in an isosceles triangle two angle bisectors to congruent sides are congruent.

3. Prove that in an isosceles triangle two altitudes to congruent sides are congruent.

4. Prove that segments of two perpendicular bisectors to two congruent sides of an isosceles triangle between midpoint of one side and crossing of its bisector with another side are congruent.
More precisely, consider an isosceles triangle ΔABC with congruent sides AB and BC. Draw perpendicular bisectors of its congruent sides through midpoint M of side AB and midpoint N of side BC. Point P is a crossing point of the perpendicular bisector of side AB with side BC. Point Q is a crossing point of the perpendicular bisector of side BC with side AB. Prove that segments MP and NQ are congruent.

5. Prove that a line, perpendicular to an angle bisector, cuts from two rays forming this angle congruent segments, assuming that an angle is less that 180 degrees.
In details, let AB and AC be two rays forming angle ∠BAC, let AM be a bisector of this angle and P - any point on this bisector. Draw a perpendicular to AM that croses it at point P. This same perpendicular crosses rays AB at point X and AC at point Y. Prove that segments AX and AY are congruent.

6. Prove that median AM of a triangle ΔABC from vertex A to an opposite side BC is equidistant from vertices B and C.
Notice, that distance from a point to a line is measured as a length of a perpendicular dropped from this point to a line.

7. Prove that the length of median AM of a triangle ΔABC from vertex A to an opposite side BC is less than half sum of lengths of sides AB and AC in lies in between.
Hint: extend the median beyond side BC by doubling its length to point D and consider triangle ΔABD.

8. Prove that sum of lengths of all three medians of a triangle is less than its perimeter, but greater than its halp-perimeter.
Hint: use the triangle inequality and the previous problem.

9. Prove that sum of lengths of two diagonals of a quadrangle is less than its perimeter, but greater than its half-perimeter.

10. Given an angle ∠PMQ and two points on each side: A and B on side MP and A' and B' on side MQ such that corresponding pairs of segments are congruent:
MA is congruent to MA' and MB is congruent to MB'.
Prove that crossing segments AB' and A'B cross on a bisector of angle ∠PMQ.
Notice, that this theorem implies an easy way to construct an angle bisector.

11. Given a sraight line PQ and two pairs of points symmetrical relative to this line: points A and A' are symmetrical relative to PQ, points B and B' are symmetrics too relative to the same line PQ.
Prove that there exists a point M on line PQ equidistant from all four points A, A', B and B'.

12. Given a straight line PQ and two points A and B on the same side from it. Find a point M on line PQ such that the sum of lengths of segments AM and MB is minimal.

13. Given an accute angle ∠PMQ and point A inside it. Find point X on side MP of this angle and point Y on its other side MQ such that a perimeter of triangle ΔAXY is minimal.