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Problem

Two ladders are placed on opposite diagonals in an alley such that one ladder reaches a units up one wall, the other ladder reaches b units up the opposite wall and they intersect h units above the ground.

Prove the following result.

Solution

Consider the following diagram:

By similar triangles:

x + y a

=

y  h

and

x + y b

=

x  h

Adding equations:

x + y a

+

x + y b

=

x + y h

Dividing by (x + y):

1   a

+

1   b

=

1   h