Two poles formula
The two pole formula/theorem states that for two poles some distance apart, the height of the intersection point of the lines joining the top of either pole to the foot of the opposite pole is half the harmonic mean of the heights of the two poles(and that this is true regardless of the distance between the two poles).
Proof using similar triangles
We are looking to show that for the two poles and of heights and , respectively, and units apart, .
Because , Because , Therefore, as desired.
Coordinate/analytical geometry proof
Using the same diagram as above, if we let be the origin, then and are the points and respectively. Hence, and have slopes of and , respectively, and y-intercepts of and , respectively. That means their equations are and , respectively. Setting the two equations equal to find the intersection point(recall that a point is shared by two lines if and only if it satisfies the equation of both lines), we obtain To find the ordinate, we plug in this to either equation of or (in this case, we choose 's equation because it contains fewer terms): so is the ordered pair . Therefore, the height is , as desired.