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Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Proof of Pick's Theorem

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Follow the five steps in the question. These steps lead you to the required proof.

(1) Here you are glueing 2 polygons along a common edge and the area of the 'sum' of the polygons is the sum of the areas of the individual polygons. Decide which lattice points on the common edges become interior lattice points to the new polygon and which are on the edge of the new polygon and make sure none of these is counted twice.

(2) Proving Pick's Theorem for a rectangle simply involves counting lattice points.

(3) Now deduce Pick's Theorem for right-angled triangles.

(4) Now you know Pick's Theorem holds for rectangles and right-angled triangles deduce that it holds for the general triangle.

(5) Finally deduce Pick's Theorem for the general plane polygon using the earlier parts of the proof.