The proof that HB equals JC utilizes congruent triangles and given proportions. Establishing congruence between triangles HKD, KFD, GFC, and AHD, as well as using the relationships between lengths AD, DF, and DG, demonstrates the required equality.
The student is asking to prove that HB = JC under the condition that HD = JD and HC = JB. By using the properties of congruent triangles, we can demonstrate the equality of HB and JC. Given the width of the Moon as seen from point H is KD = x, and that the Moon's and Sun's apparent sizes are about the same, angle KHD is 0.5 degrees. The extension of line AD = R to a further distance R arriving at point F implies that triangles HKD and KFD are congruent due to their shared side HD (equal to JD by given), their right angles at D, and their equal 0.5-degree angles at H and F respectively. Consequently, triangles GFC and AHD are also congruent to the shaded triangles HKD and KFD because they share corresponding angles and sides.
Therefore, since AH = GF (by congruency of triangles AHD and GFC) and HD = DF, it follows that AD = DG (both equal to R). Moreover, AB = 3x by the proportions given, and since AC = AD + DC and DC = DG + GC, we can substitute to find AC = 3R, paralleling the proportionality with AB = 3x. In summary, the given elements and the congruency of the involved triangles ensure that HB = JC.