Proofs in plane geometry：練習問題

In \(\triangle ABC\), \(D\) and \(E\) are points on \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If the ratio \(AD : DB = 2 : 3\) and the area of \(\triangle ADE\) is 8 cm\(^2\), find the area of the quadrilateral \(DBCE\).

From an external point \(P\), a tangent \(PT\) and a secant line \(PAB\) are drawn to a circle, where \(A\) and \(B\) are points on the circumference. If \(PT = 12\) cm and \(PA = 8\) cm, find the length of the chord \(AB\).

In a circle, two chords \(AB\) and \(CD\) intersect at an interior point \(X\). If \(AX = 6\) cm, \(XB = 4\) cm, and \(CX = 3\) cm, calculate the length of \(XD\).

\(ABCD\) is a cyclic quadrilateral where \(AB\) is the diameter of the circle. Given that \(AD = DC\) and \(\angle BAC = 20^\circ\), calculate the magnitude of \(\angle DAC\).

\(PAT\) is a tangent to a circle at point \(A\). \(B\) and \(C\) are points on the circumference such that \(C\) is in the major segment. If \(\angle TAB = 72^\circ\), find the value of \(\angle ACB\) by using the tangent-chord theorem.

In a circle with center \(O\), \(AB\) is a diameter and \(AC\) is a chord. If \(M\) is the midpoint of \(AC\), prove that \(OM\) is parallel to the chord \(BC\).

From an external point \(P\), a tangent \(PT\) and a secant \(PAB\) are drawn to a circle, where \(A\) and \(B\) lie on the circumference. Prove that \(\triangle PTA\) is similar to \(\triangle PBT\) and deduce that \(PT^2 = PA \cdot PB\).

In a circle, chords \(AB\) and \(CD\) intersect at a point \(X\) inside the circle. Prove that \(\triangle AXD\) is similar to \(\triangle CXB\).

In the diagram, P, Q, and R are points on the circumference of a circle. The line PT is a tangent to the circle at point P. A line through Q is drawn parallel to the tangent PT, and it intersects the chord PR at point S.

(a) Prove that triangle PQS is similar to triangle PRP is incorrect. Prove that \(\triangle PQS\) is similar to \(\triangle RQP\).
(b) Hence, show that \(PQ^2 = PS \times PR\).

In the diagram, two circles intersect at points \(A\) and \(B\). A common tangent touches the first circle at \(P\) and the second circle at \(Q\).
(a) Prove that \(∠ PAB + ∠ QAB = 180^∘ - ∠ PAQ\).
(b) By considering the properties of tangents and chords, prove that \(∠ PAQ + ∠ PBQ = 180^∘\).