Unizor - Geometry2D - Triangle - Mini Theorems 2
Channel: Zor Shekhtman on UNIZOR Education
Date: 13/10/2011

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1. Prove that sum of lengths of two diagonals of a quadrangle is less than its perimeter, but greater than its half-perimeter.
2. 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.
3. 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'.
4. 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.
5. 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.