# G(幾何)

## 圓

## 範例$1.2.$
Given an isosceles triangle $ABC$ with $AB = AC$. The midpoint of side $BC$ is denoted by $M$. Let $X$ be a variable point on the shorter arc $MA$ of the circumcircle of triangle $ABM$. Let $T$ be the point in the angle domain $BMA$, for which $\angle TMX = 90∘ $ and $TX = BX$. Prove that $\angle MTB − \angle CTM$ does not depend on $X$. (2007 ISL G2)

## 範例$1.3.$
Let ABCD be a trapezoid with parallel sides AB > CD. Points K and L lie on the line segments AB and CD, respectively, so that AK/KB = DL/LC. Suppose that there are points P and Q on the line segment KL satisfying
$\angle APB = \angle BCD$ and $\angle CQD = \angle ABC$. Prove that the points $P, Q, B$ and $C$ are concyclic. (2006 ISL G2)

## 範例$2.2.$
令 $I$ 為三角形 $ABC$ 的內心，點 $P$ 在三角形的內部，滿足
$\angle PBA + \angle PCA = \angle PBC + \angle PCB$.
試證：$AP≥ AI$，且等號成立的充份必要條件為 $P=I$. (2006 IMO P1)

## 範例$2.3.$
設圓內接四邊形$ABCD$的外接圓為$\omega$，半徑為$r$且對角線$AC$和$BD$相交於$P$。假設$AD=DP$，$S$為從$P$到$AB$的垂足點而點$Q$位於直線$SP$上，使得$PQ=r$且$S,P,Q$依序位於直線上。令通過$A$且垂直CQ的直線與通過$B$垂直於$DQ$的直線相交於$E$，證明$E$位於$\omega$上 (2024 Taiwan TST Round 1 Mock Exam P1)