Proving that FR = CR will do it, because you will be able to show that the triangles OFR and OCR (where O is the centre of the circle) are congruent, implying that the angles CRO and FRO are equal.
(You can then apply the same argument to the other two corners of the triangle).
So, draw a new diagram containing just the circle and the line segments FER and CDR. Draw lines from D to F and from C to E. Prove that the triangles CER and FDR are similar. (Angle DFE = angle DCE and angle DRE is common to both triangles).
Write down the ratios of the two triangles, FR/CR = DR/ER. Cross multiply and use the fact that FR=FE + ER = m + ER, and CR = CD + DR = m + DR, (where m is the common length stated in the question).
Now, after a little algebra, you can show that DR = ER, implying that FR = CR, etc..
