접선, 할선, 원주각 종합 고난도 문제

문제

그림과 같이 원에 내접하는 사각형 $ABCD$가 있습니다. 점 $A$를 지나는 직선 $l$은 원에 접하고, 점 $E$는 직선 $l$ 위에 있습니다. 직선 $BC$를 연장한 선과 직선 $l$은 점 $F$에서 만납니다. 주어져 있는 각의 크기는 다음과 같습니다.

$\angle EAD = 60^\circ$ $\angle CAD = 45^\circ$ $\angle BFC = 20^\circ$

이때, $\angle BCD$의 크기를 구하시오.

<!-- Circle (Cyclic Quadrilateral) -->
<circle cx="200" cy="150" r="100" class="line"/>

<!-- Points A, B, C, D -->
<!-- A(200, 50) -->
<!-- D(286.6, 100) (approx for 60 degrees tangent-chord) -->
<!-- F(300, 50) C(254.8, 66.3) B(104.4, 120.4) (approx for 20 degrees BFC) -->
<circle cx="200" cy="50" r="3" class="point" id="A"/>
<text x="200" y="40" class="label">A</text>
<circle cx="104.4" cy="120.4" r="3" class="point" id="B"/>
<text x="90" y="125" class="label">B</text>
<circle cx="254.8" cy="66.3" r="3" class="point" id="C"/>
<text x="265" y="60" class="label">C</text>
<circle cx="286.6" cy="100" r="3" class="point" id="D"/>
<text x="295" y="105" class="label">D</text>

<!-- Tangent line l and points E, F -->
<line x1="0" y1="50" x2="400" y2="50" class="line"/>
<circle cx="100" cy="50" r="3" class="point" id="E"/>
<text x="100" y="40" class="label">E</text>
<circle cx="300" cy="50" r="3" class="point" id="F"/>
<text x="300" y="40" class="label">F</text>

<!-- Chords of the cyclic quadrilateral -->
<line x1="200" y1="50" x2="104.4" y2="120.4" class="line"/> <!-- AB -->
<line x1="104.4" y1="120.4" x2="254.8" y2="66.3" class="line"/> <!-- BC -->
<line x1="254.8" y1="66.3" x2="286.6" y2="100" class="line"/> <!-- CD -->
<line x1="286.6" y1="100" x2="200" y2="50" class="line"/> <!-- DA -->

<!-- Diagonals used for angles -->
<line x1="200" y1="50" x2="254.8" y2="66.3" class="dashed-line"/> <!-- AC -->
<line x1="104.4" y1="120.4" x2="286.6" y2="100" class="dashed-line"/> <!-- BD -->

<!-- Extended line FBC -->
<line x1="300" y1="50" x2="50" y2="150" class="line"/> <!-- F-C-B extended. F(300,50), C(254.8,66.3), B(104.4,120.4). Need to draw full line. -->
<line x1="300" y1="50" x2="30" y2="182" class="line"/>

<!-- Angle EAD = 60 degrees -->
<path d="M 150 50 A 50 50 0 0 1 175 78.86" class="angle-arc"/>
<text x="155" y="65" class="angle-label">60°</text>

<!-- Angle CAD = 45 degrees -->
<path d="M 230 70 A 30 30 0 0 1 236.6 86.6" class="angle-arc"/>
<text x="240" y="75" class="angle-label">45°</text>

<!-- Angle BFC = 20 degrees -->
<path d="M 280 50 A 20 20 0 0 0 286.84 57.08" class="angle-arc"/>
<text x="275" y="65" class="angle-label">20°</text>