Problems in Geometry Series 2

I have uploaded a new lecture to http://www.unizor.com/ with the following problems.

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 ΔABC. Segment AD bisects angle ∠BAC (point D lies on side BC). Straight line through point D is parallel to side AC, intersecting side AB at point E. Straight line through point E is parallel to side BC, intersecting side AC at point F. Prove that segments AE and CF are congruent.

5. Given an angle ∠MXN. Inside it an angle ∠PYQ is positioned in such a way that MX is parallel to PY, NX is parallel to QY and distance between MX and PY is equal to distance between NX and QY. Prove that a bisector of angle ∠MXY is a bisector of angle ∠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 ΔABC. Let vertex B be at the top and side AC be a base. Let point X be an intersection of two bisectors of angles at the base. Straight line through point X parallel to a base intersects its left side AB and right side BC at points M and N correspondingly. Prove that segment MN is equal to a sum of segments AM and 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.