### 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:

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. (In the problem 'Pareq Calc' the existence of the equilateral triangle was assumed.)

If you have Java enabled, it may help to use the dynamic diagram below which shows three parallel lines and the fixed point A on one of the lines. The points B and C are free to move along the other two parallel lines in such a way that the lengths AB and AC are always equal.

Click and drag the red points.