### Three Way Split

Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.

### Pareq Calc

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.

### Farhan's Poor Square

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

# Pareq Exists

##### Stage: 4 Challenge Level:

This is another tough nut mind bender, and we have to give a proof of existence. The conjecture is that, given any set of three parallel lines, there always exists an equilateral triangle with one of its vertices on each of the lines. The trouble with this sort of mathematics is that there are infinitely many possible cases. It means nothing that the conjecture holds true in every case we test because it might still break down in a case we have not tried. So what can be done? Playing with the dynamic diagram suggests that the length of BC changes - how does it change?