Trying to understand the maths of a church window
Hello everyone !
Currently, I am drawing a lot of churches (and other religious buildings, mostly european and christian) windows and arcades for fun.
I came upon the following "window" (see image attached) (it's from Mosteiro dos Jerónimos near Lisboa in Portugal) and I was wondering, how can we calculate the diameter of the small circle at the top, knowing the diameter of the big circle that hosts it ?
I could not find myself, so I asked my friend who is math savvy and they helped me find a way to calculate it.
If you look at the graph they made (basically the window is the space between C, B and G, and the small circle is the circle of center I and of diameter GI), I will explain how we did it.
We consider the square triangle FCI.
According the Pythagore, we know that
FC² + IF² = CI²
Let's write the big circle's radius d (which is FC), and the small circle radius x (which is IG).
We know that FC = d, IF = d - x, and CI = d + x
So we have the following expression
d² + (d - x)² = (d + x)²
d² + d² - 2*d*x + x² = d² + 2*d*x + x²
d² = 4*d*x
d/4 = x.
So there is a simple 4 factor between the two diameters !
I hope you'll find it interesting. bye bye