### 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.

### Pareq Exists

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

# Farhan's Poor Square

##### Stage: 4 Challenge Level:

Splendid solutions to Farhan's problem came in from Shabbir Telani, age 13, Jack Hunt School, Peterborough, Prav Idaikkadar,age 13, and Megan Mitchell, age 14, North London Collegiate School Maths Club, Richard Mason, Madras College, and Rachel Evans, the Mount School York. This is Shabbir's solution:

Using trigonometry:

$$x = \frac{35}{\cos 42^o} = 47.09714 = 47.10$$ (to 2 decimal places).

To find the area of the square not covered by the circle work out the area of the square minus the area of the circle.

The formula for the area is

$A = x^2 - \pi \left(\frac{x}{2}\right)^2$

So the area of the square not covered by the circle and triangle is 2218.14 - 1742.12 = 476.02 cm 2 (to 2 decimal places).