If a point is in the interior of triangle distance and from the sides, what is the ratio of the area of quadrilateral to that of ?
One way of determining this is to draw parallels to the sides of the triangles through . Let and be where these parallels meet side as shown below.
Let the sides of the triangles have lengths with corresponding altitudes .
Then as and are similar,
where the second last line follows from twice the area of |ABC| being .
Combining (1), (2) and (3), we obtain our desired answer as
Similar formulas can be found for quadrilaterals and by permuting variables. Note that if is outside the triangle or if the triangle is obtuse-angled, care must be taken in the signs of the areas (the quadrilaterals may not be convex) and variables .
Note that (4) may also be written as
1) If is equilateral, and from (4) we obtain
2) If is at the incentre of , then (the inradius) and from (4) we have
3) If is right-angled at , then quadrilateral is a rectangle with area and has area and from (5),
4) If and (symmetric isosceles triangle case) then from (4),