### Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

### Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

### No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

# Half a Triangle

##### Stage: 4 Challenge Level:

If the area ratio has to be $1:2$ , what would the line ratio need to be?

What constructions do you know that create such a ratio?

Can you see how to create such a construction on the given triangle?