Given a square ABCD with unknown side.
A point P in the plane of ABCD has distances a, b, c from the corners A, B and D respectively.
Determine the size of the square in terms of a, b and c.

Solution:

Method 1
By Trigonometry
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Method 2
Application of Coordinates GeometryClick to reveal hidden text
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Method 3
Rotation of Figure
Assume the case with P inside ABCD.
Rotate triangle PAB around the corner A through 90 degrees in the counter-clockwise sense.
Then P will be rotated to a new point, say Q, with AQ = a and DQ = b.
Calculating the area of APDQ in two different ways:
First: Sum of areas of triangles APQ and PDQ by Heron's Formula in Expanded form.
Second: Area of quadrilateral APDQ by Bretschneider's Formula.