The point O lies inside the triangle EFG, and ∠OEF = ∠OFG = ∠OGE.

Given that EF = 13, FG = 14, and GE =15, determine tan ∠OEF

For a problem asking about a tangent value, it took a lot of sines to get there!

Let T be the common measure of angles OEF, OFG, and OGE. The problem asks us to find tan(T).

Start with triangle EFG. By Heron's Formula the area is:

s = (13+14+15)/2 = 21

Area = sqrt[21*(21-13)*(21-14)*(21-15)] = 84.

From the sine area formula the area equals:

(1/2)*13*14*sin(EFG) = 91*sin(EFG)

(1/2)*13*15*sin(EFG) = (195/2)*sin(FEG)

(1/2)*14*15*sin(EFG) = 105*sin(EGF)

Equating each sine area to 84 yields:

sin(EFG) = 12/13

sin(FEG) = 56/65

sin(EGF) = 4/5

Looking at the interior angles gives:

angle OGF = angle EGF - T

angle OEG = angle FEG - T

angle OFE = angle GFE - T

angle FOG = pi - angle EGF

angle EOG = pi - angle FEG

angle EOF = pi - angle GFE

Applying the law of sines to triangles EOF, EOG, amd FOG yields:

sin(EOF)/13 = sin(T)/FO

sin(FOG)/14 = sin(T)/GO

sin(EOG)/15 = sin(T)/EO

Rearranging to solve for interior edges:

FO = (169/12)*sin(T)

GO = (35/2)*sin(T)

EO = (975/56)*sin(T)

Then use the sine area formula on all three interior triangles EOF, EOG, and FOG:

area EOF = (1/2)*EO*EF*sin(T) = (12675/112)*(sin(T)^2)

area FOG = (1/2)*FO*FG*sin(T) = (1183/12)*(sin(T)^2)

area EOG = (1/2)*GO*EG*sin(T) = (525/4)*(sin(T)^2)

Add these up and equate to the known total area and solve:

(12675/112)*(sin(T)^2) + (1183/12)*(sin(T)^2) + (525/4)*(sin(T)^2) = 84

(115249/336)*(sin(T)^2) = 84

sin(T) = 168/sqrt(115249)

Then cos(T) = 295/sqrt(115249)

Finally the answer tan(T) = 168/295 = 0.5694915254, which matches Jer's numeric approximation.

The expression for sin(T) turns out to be:

2*area(EFG) / sqrt[(EF*EG)^2 + (EF*FG)^2 + (EG*FG)^2]

Similarily, the expression for cos(T):

(1/2)*(EF^2+EG^2+FG^2) / sqrt[(EF*EG)^2 + (EF*FG)^2 + (EG*FG)^2]

And tan(T) = 4*area(EFG)/(EF^2+EG^2+FG^2)

*Edited on ***January 11, 2016, 1:07 pm**