### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

### Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

### Fermat's Poser

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

# Inscribed Semicircle

##### Age 14 to 16 Short Challenge Level:

The diagram below shows a right angled triangle with sides of length $5\text{cm}$, $12\text{cm}$ and $13\text{cm}$. What is the radius of the inscribed semicircle whose base lies on the side of length $12\text{cm}$?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.