So angle MPA= angle MPC=90 degrees.
Because the line segment CM rotates 90 degrees clockwise around point C to get CN.
So MCN angle = BCG angle =90 degrees.
CM=CN
Because angle ACB= angle BCG+ angle ACM=90 degrees.
So angle ACM= angle BCN
Because AC=BC
So triangle ACM congruent triangles BCN
So angle EAC= angle DBC
Because the angle EAC= 15 degrees
So EBC angle = 15 degrees.
Because the angle ACB=90 degrees
AC=BC
So the triangle ACB is a right triangle.
So angle BAC= angle ABC=45 degrees.
AB^2=AC^2+BC^2
Because angle ABD= angle ABC+ angle EBC=45+ 15=60 degrees.
So angle BAD= angle BAC- angle EAC=45- 15=30 degrees.
Because angle BAD+ angle ABD+ angle ADB= 180 degrees.
So the angle ADB=90 degrees
So the triangle ADB is a right triangle.
So BD= 12/AB.
AB^2=AD^2+BD^2
Because BD=a
So AB=2a
AD= root 3a
AC=BC= root number 2a
So BD/AC= 1/ root number 2.
Because angle ACB= angle ADB=90 degrees (proved)
Angle AOC= angle BOD (equal to vertex angle)
So BD/AC=OB/OA=OD/OC= 1/ root number 2.
So OC = root 2OD.
In the right triangle BDO
OB^2=BD^2+OD^2
OB=BC-OC= root number 2a- root number 20d
So OD=(2- root number 3)a/2.
OC=(2 times the square root of the radical number 2-6) a/2
Because angle CMO= angle ACM+ angle EAC
Angle ACM=60 degrees
So CMO angle =60+ 15=75 degrees.
Because angle ACB+ angle EAC+ angle AOC= 180 degrees.
So the angle AOC=75 degrees
So AOC= angle CMO=75 degrees.
So MC=OC=(2 times the square root of the root number 2-6) a/2.
Because angle MPC+ angle ACM+ angle CMP= 180 degrees.
So the angle CMP=30 degrees
So CP= 1/2CM=(2 times the square root of the root number 2-6) a/4.
PM= root number 3/2CM=(2 times the square root of 6 -3 times the root number 2)a/4
Because AM=AC-CM=(2 times the square root of the root number 2+6) a/2.
In right triangle APM, the angle MPA=90 degrees.
At am 2 = pm 2+am 2
So AM=(6-2 times the root number 3) a.
Because DM=AD-AM
So DM=(3 times the root number 3-6) a.