1. Prove that a median to a hypotenuse in the right triangle is equal to half of that hypotenuse.

Hint: extend the median through hypotenuse by its length.

2. Prove a converse theorem, that if a median in a triangle is equal to half a side it falls on, then an angle, where it starts, is the right angle.

3. Prove that in a right triangle a median and an altitude to its hypotenuse form an angle, equal to difference between this triangle's acute angles.

Hint: use a previous theorem.

4. Given triangle Δ

**. Segment**

*ABC***bisects angle ∠**

*AD***(point**

*BAC***lies on side**

*D***). Straight line through point**

*BC***is parallel to side**

*D***, intersecting side**

*AC***at point**

*AB***. Straight line through point**

*E***is parallel to side**

*E***, intersecting side**

*BC***at point**

*AC***. Prove that segments**

*F***and**

*AE***are congruent.**

*CF*5. Given an angle ∠

**. Inside it an angle ∠**

*MXN***is positioned in such a way that**

*PYQ***is parallel to**

*MX***,**

*PY***is parallel to**

*NX***and distance between**

*QY***and**

*MX***is equal to distance between**

*PY***and**

*NX***. Prove that a bisector of angle ∠**

*QY***is a bisector of angle ∠**

*MXY***.**

*PYQ*6. Prove that any segment, that connects two bases of a trapezoid, is divided by a median in two congruent parts.

7. Given a triangle Δ

**. Let vertex**

*ABC***be at the top and side**

*B***be a base. Let point**

*AC***be an intersection of two bisectors of angles at the base. Straight line through point**

*X***parallel to a base intersects its left side**

*X***and right side**

*AB***at points**

*BC***and**

*M***correspondingly. Prove that segment**

*N***is equal to a sum of segments**

*MN***and**

*AM***.**

*CN*8. Straight lines are drawn through all three vertices of a triangle, forming another (bigger) triangle. Prove that this bigger triangle is divided by sides of a smaller one into 4 triangles, each congruent to a small triangle and each side of a bigger triangle is twice as big as parallel to it side of a smaller triangle.

9. Prove that in an isosceles triangle sum of two distances from any point on a base to two legs is constant and is equal to an altitude from any of two end points of a base to a leg.

10. Change a condition of a previous theorem to use a point on continuation of a base (outside of a triangle). Formulate a theorem in this case and prove it.

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