In the above figure, seg PA, seg QB and seg RC are perpendicular to seg AC. From the information given in the figure, prove that:
`(1)/(x) + (1)/(z) = (1)/(y)`

Explanation

In `Delta PAC` and `Delta QBC`
`/_PAC ~= /_ QBC`….(given)
`/_ PCA ~= /_QCB`….(Common angle)
`:. Delta PAC ~ Delta QBC` …..(By AA test)
`:. (PA)/(QB) = (AC)/(BC)`….(Corresponding sides of similar tirangles are in proportion)
`(x)/(y) = (AC)/(BC)`
`(y)/(x) = (BC)/(AC)`...(I) ...(by invertendo)
Similarly, `Delta RCA ~ Delta QBA`..... (By AA test)
`(RC)/(QB) = (AC)/(AB)`...(Corresponding sides of similar triangles are in proportion
`:. (z)/(y) = (AC)/(AB)`
`:. (y)/(z) = (AB)/(AC)`.....(II)....(by invertendo)
`:. (y)/(x) + (y)/(z) = (BC)/(AC) + (AB)/(AC)` [Adding (I) and (II)]
`:. y ((1)/(x) + (1)/(Z)) = (BC + AB)/(AC)`
`:. y ((1)/(x) + (1)/(z)) = (AC)/(AC) (A-B-C)`
`:. ((1)/(x) + (1)/(z)) = 1`
`:. (1)/(x) + (1)/(z) = (1)/(y)`