[Math Lair] Pillow-Problems: Problem #12

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For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

12.

Given the semi-perimeter and the area of a Triangle, and also the volume of the cuboid whose edges are equal to the sides of the Triangle: find the sum of the squares of its sides.

[23/1/91

Answer:

12.

If s = semi-perimeter, m = area, v = volume; then

a² + b² + c² = 2 · (s² − vsm²s².

Solution:

12.

Let s = semi-perimeter, m = area, v = volume.

We know that m = s · (sa) · (sb) · (sc);

m² = s · (sa) · (sb) · (sc);

m²
s
= s³ − s²·(a + b + c) + s · (bc + ca + ab) − abc;

= s³ - 2s² + s · (bc + ca + ab) − v;

m²
s²
+
v
s
+ s² = bc + ca + ab;

∴ 2 ·(
m²
s²
+
v
s
+ s²) = (a + b + c)² − (a² − b² + c²);

= 4s² − (a² − b² + c²);

a² + b² + c² = 2 · (s² −
v
s
m²
s²
).