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 Δ

**with congruent sides**

*ABC***and**

*AB***. Draw perpendicular bisectors of its congruent sides through midpoint**

*BC***of side**

*M***and midpoint**

*AB***of side**

*N***. Point**

*BC***is a crossing point of the perpendicular bisector of side**

*P***with side**

*AB***. Point**

*BC***is a crossing point of the perpendicular bisector of side**

*Q***with side**

*BC***. Prove that segments**

*AB***and**

*MP***are congruent.**

*NQ*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

**and**

*AB***be two rays forming angle ∠**

*AC***, let**

*BAC***be a bisector of this angle and**

*AM***- any point on this bisector. Draw a perpendicular to**

*P***that croses it at point**

*AM***. This same perpendicular crosses rays**

*P***at point**

*AB***and**

*X***at point**

*AC***. Prove that segments**

*Y***and**

*AX***are congruent.**

*AY*6. Prove that median

**of a triangle Δ**

*AM***from vertex**

*ABC***to an opposite side**

*A***is equidistant from vertices**

*BC***and**

*B***.**

*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

**of a triangle Δ**

*AM***from vertex**

*ABC***to an opposite side**

*A***is less than half sum of lengths of sides**

*BC***and**

*AB***in lies in between.**

*AC*Hint: extend the median beyond side

**by doubling its length to point**

*BC***and consider triangle Δ**

*D***.**

*ABD*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 ∠

**and two points on each side:**

*PMQ***and**

*A***on side**

*B***and**

*MP***and**

*A'***on side**

*B'***such that corresponding pairs of segments are congruent:**

*MQ***is congruent to**

*MA***and**

*MA'***is congruent to**

*MB***.**

*MB'*Prove that crossing segments

**and**

*AB'***cross on a bisector of angle ∠**

*A'B***.**

*PMQ*Notice, that this theorem implies an easy way to construct an angle bisector.

11. Given a sraight line

**and two pairs of points symmetrical relative to this line: points**

*PQ***and**

*A***are symmetrical relative to**

*A'***, points**

*PQ***and**

*B***are symmetrics too relative to the same line**

*B'***.**

*PQ*Prove that there exists a point

**on line**

*M***equidistant from all four points**

*PQ***,**

*A***,**

*A'***and**

*B***.**

*B'*12. Given a straight line

**and two points**

*PQ***and**

*A***on the same side from it. Find a point**

*B***on line**

*M***such that the sum of lengths of segments**

*PQ***and**

*AM***is minimal.**

*MB*13. Given an accute angle ∠

**and point**

*PMQ***inside it. Find point**

*A***on side**

*X***of this angle and point**

*MP***on its other side**

*Y***such that a perimeter of triangle Δ**

*MQ***is minimal.**

*AXY*
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