Two pillars of height $a$ and $b$ subtend the same angle $\alpha$ at a point on the line joining their feet. If the pillars subtend angles $\beta$ and $\gamma$ at another point in the horizontal plane at which the line joining their feet subtends a right angle, then ${ \left( { a+b } \right) }^{ 2 }\cot ^{ 2 }{ \alpha } ={ a }^{ 2 }\cot ^{ 2 }{ \beta } +{ b }^{ 2 }\cot ^{ 2 }{ \gamma }.$

At the point on the line joining the feet, the distance from that point to the base of the pillars is (a + b) / (2 tan alpha), making (a + b)^2 cot^2 alpha equal to 4 times the square of this distance. At the second point in the horizontal plane, the squared distances to the bases of the pillars are a^2 cot^2 beta and b^2 cot^2 gamma because the lines joining the feet subtend a right angle. By the Pythagorean theorem, the sum of these two squared distances equals four times the squared distance from the first point, proving the equation is True.

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