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Consider Right tri'gle HFC. We have to figure out if R is mid-point of OG.

Two right tri'gles HFC and HOR are similar triangles by angle-side-angle (asa) property.

1) Angle RHO is same as Angle CHF.

2) HO and HF are in straight line, given that O is mid-point of HF, HO/HF = 1/2.

3) Angle HOR is same as Angle HFC = 90 degree.

So HO/HF = HR / HC = OR / FC = 1/2.

Hence, OR is half of FC. And OG = FC = 1. Therefore, R is mid-point of OG.

You can take it from here.

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