# Product of Areas of Three Dissimilar Right Triangles

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BobDH
Product of Areas of Three Dissimilar Right Triangles

The formula below will find the product P for the areas of 3 right triangles A, B & C, as described in Geometry Post, Three Dissimilar Right Triangles.

P = (ab)^4 / 2c^2

1. Triangle C is a known right triangle, with legs "a" & "b", and hypotenuse "c".

2. a is the short leg and b is the long leg of C

3. Hypotenuse of A = 2(b) of C.

4. Hypotenuse of B = 2(a) of C.

5. Altitudes to hypotenuse in A & B
are identical to that in C.

EXAMPLE:
Pythagorean triangle 3 4 5 is selected. a = 3, b = 4, c = 5.
1a. Altitude to hypotenuse in C is 3 x 4 /5 = 2.40.

2a. Hypotenuse in A is 2(4) = 8.00

3a. Hypotenuse in B is 2(3) = 6.00

4a. Area of A = 8 x 2.40 / 2 = 9.60.

5a. Area of B = 6 x 2.40 / 2 = 7.20.

6a. Area of C = 3 x 4/2 = 6.00.

7a. Prod of Areas of A, B, C = 9.60 x 7.20 x 6.00 = 414.720.

8a. Using the formula P = (ab)^4 / 2c^2; P = (3x4)^4 / 2x5^2 = 20,736.000 / 50 = 414.720.

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