Equilateral Triangle Within a Square - Problem With Solution
A problem on the proof of an equilateral triangle within a square is presented along with detailed solution.
In the figure below, ABCD is a square. Prove that triangle DMC is equilateral.
Solution to Problem :
AMB is an isosceles triangle and therefore MA = MB. Also the sizes of angles MAD and MBC are equal. Hence triangles DAM and CBM have an equal angle between two equal sides are therefore congruent. Since triangles DAM and CBM are congruent sides DM and CM are equal in size and therefore triangle DCM is isosceles.
We draw the perpendicular MM' to CB and we consider the right triangles MCM' and MBM' and the tangent of angles MCM' and MBM'.
tan(MCM') = MM' / CM'
tan(MBM') = MM' / BM'
The above may be written as
CM' = MM' / tan(MCM') and BM' = MM' / tan(MBM')
Note that the size of angle MBM' is equal to 75 degrees. Let the length of the side of the square be x. Hence
MM' = x / 2 and CM' + BM' = x
Substitute the above in the equations CM' = MM' / tan(MCM') and BM' = MM' / tan(MBM') to obtain
x = (x/2) / tan(MCM') + (x/2) / tan(75 degrees)
Divide all terms of the equation by x and multiply them by 2 to obtain
2 = 1 / tan(MCM') + 1 / tan(75 degrees)
tan(75 degrees) is calculated using the tangent of a sum formula follows
tan(75 degrees) = tan(30 degrees + 45 degrees) =
[ tan(30) + tan(45) ] / [ 1 - tan(30)*(tan(45) ] = 2 + sqrt(3)
Substitute tan(75 degrees) by 2 + sqrt(3) into the equation 2 = 1 / tan(MCM') + 1 / tan(75 degrees) to obtain
1 / tan(MCM') = 2 - 1 / [ 2 + sqrt(3) ]
Simplify the above and solve for tan(MCM') to obtain
tan(MCM') = 1 / sqrt(3)
Solve the above trigonometric eqution to find the size of angle MCM'
MCM' = 30 degrees
The size of angle DCM is now calculated as follows
DCM= 90 degrees - 30 degrees = 60 degrees
and since triangle DCM is isosceles then angle CDM is also equal to 60 degrees and the triangle is equilateral.