求救2題資優班考古題
如右圖,在\(\Delta ABC\)中,\(D\)為線段\(\overline{AC}\)上的一點,且\(\overline{BD}\)長為\(\overline{AD}\)長的\(\sqrt{2}\)倍,\(E\)是\(\overline{BD}\)的中點,\(F\)是直線\(EA\)上一點,連接\(\overline{DF}\),並在\(\overline{DF}\)上取一點\(G\),使得\(\angle AGF=\angle CAE\),在直線\(BG\)上取一點\(H\),使得\(\overline{CH}// \overline{AB}\)。試證:
(1)\(\Delta ABD\)與\(\Delta EAD\)相似。
(2)\(C\)、\(D\)、\(G\)、\(H\)四點共圓。
\(\overline{PA},\overline{PB}\)是圓\(O\)的切線\(\overline{BE}// \overline{PD}\),\(\overline{PD}\)交\(\overline{AE}\)於點\(M\),試證:\(M\)是\(\overline{CD}\)中點。
拜託各位老師了,謝謝 1.
\(\displaystyle \angle{AGF}=\angle{CAE}=\angle{ABD} \)
故 \( A、B、D、G \) 四點共圓
於是 \(\displaystyle \angle{BGD}=\angle{BAD}=\angle{DCH} \)
所以 \( C、D、G、H \) 四點共圓。
2.
\(\displaystyle \angle{AMP}=\angle{AEB}=\frac{1}{2} \overset{\frown}{AB}=\angle{AOP} \)
故 \( A、M、O、P \) 四點共圓
\(\displaystyle \angle{OMP}=\angle{OAP}=90^o \)
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