MathLabs

Problem 4

Let ABC be an acute triangle. Let D be the foot of the perpendicular from A to BC, M the midpoint of BC, and H the orthocenter of ABC. Let E be the intersection of the circumcircle Gamma of ABC with the ray MH, and let F be the other intersection of line ED with Gamma. Prove that BF/CF = AB/AC, where XY denotes the length of segment XY.
Step 2 of 6: Use the antipodal point
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K∈Γ,AK is a diameter,∠BCK=∠CBH,∠CBK=∠BCH.K\in\Gamma,\quad AK\text{ is a diameter},\quad \angle BCK=\angle CBH,\quad \angle CBK=\angle BCH.
Detailed analysis

Let K be the point on Gamma for which AK is a diameter. Since angles AKB and AKC are right, the acute-triangle angle relations give angle BCK=angle CBH and angle CBK=angle BCH.