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a) Ta có $\widehat{CMD}=\widehat{CAD}=45^\circ$ (góc nội tiếp cùng chắn cung CD) $\widehat{CDA}=90^\circ$ nên $\widehat{CMD}+\widehat{DMC}=90^\circ$ $\Rightarrow \widehat{DMC}=45^\circ$ $\Rightarrow \widehat{CMD}=\widehat{DMC}\Rightarrow MD=CD$ $\Rightarrow \Delta MDC$ cân tại M $\Rightarrow \widehat{MDC}=\widehat{MCD}=45^\circ$ $\Rightarrow \widehat{BMC}=45^\circ$ $\Rightarrow \widehat{BCM}=90^\circ$ $\Rightarrow BM=BC$ $\Rightarrow \Delta BMC$ cân tại M $\Rightarrow \widehat{CBM}=\widehat{BCM}=45^\circ$ $\Rightarrow \widehat{ABM}=45^\circ$ $\Rightarrow \Delta ABM$ cân tại M $\Rightarrow AM=BM$ $\Rightarrow \Delta AMD=\Delta BMC(c.c.c)$ $\Rightarrow DM=MC$ $\Rightarrow DM$ đi qua trung điểm của đoạn BC. b) Ta có $\widehat{ECD}=45^\circ$ (góc nội tiếp cùng chắn cung ED) $\Rightarrow \widehat{ICE}=45^\circ$ $\Rightarrow \widehat{ICK}=45^\circ$ $\Rightarrow \widehat{CKD}=45^\circ$ $\Rightarrow \widehat{CKD}=\widehat{KCD}$ $\Rightarrow CK=KD$ $\Rightarrow \Delta CKD$ cân tại K $\Rightarrow \widehat{KDC}=\widehat{KCD}=45^\circ$ $\Rightarrow \widehat{KDI}=45^\circ$ $\Rightarrow \widehat{KID}=45^\circ$ $\Rightarrow \widehat{KID}=\widehat{KDI}$ $\Rightarrow IK=DK$ $\Rightarrow DK=\frac{1}{2}AD$ $\Rightarrow DK=\frac{1}{2}a$ $\Rightarrow K$ cố định $\Rightarrow F$ cố định c) Ta có $\widehat{ACD}=45^\circ$ (góc nội tiếp cùng chắn cung AD) $\Rightarrow \widehat{ACI}=45^\circ$ $\Rightarrow \widehat{CAI}=45^\circ$ $\Rightarrow \widehat{CAI}=\widehat{ACI}$ $\Rightarrow AI=CI$ $\Rightarrow \Delta CAI$ cân tại C $\Rightarrow \widehat{AIC}=90^\circ$ $\Rightarrow \widehat{CID}=90^\circ$ $\Rightarrow \widehat{CID}+\widehat{KCD}=180^\circ$ $\Rightarrow CI//KD$ $\Rightarrow \frac{AI}{ID}=\frac{CI}{KD}$ $\Rightarrow \frac{AI}{ID}=\frac{CI}{\frac{1}{2}a}$ $\Rightarrow \frac{AI}{ID}=\frac{2CI}{a}$ $\Rightarrow \frac{AI}{ID}=\frac{2CE}{ED}$ $\Rightarrow \frac{AI}{ID}=\frac{2(a-x)}{x}$ $\Rightarrow \frac{AI}{ID}=\frac{2a-2x}{x}$ $\Rightarrow \frac{AI}{ID}=\frac{2a}{x}-2$ $\Rightarrow \frac{AI}{ID}+2=\frac{2a}{x}$ $\Rightarrow \frac{AI+2ID}{ID}=\frac{2a}{x}$ $\Rightarrow \frac{AD+ID}{ID}=\frac{2a}{x}$ $\Rightarrow \frac{a+ID}{ID}=\frac{2a}{x}$ $\Rightarrow \frac{a}{ID}+1=\frac{2a}{x}$ $\Rightarrow \frac{a}{ID}=\frac{2a}{x}-1$ $\Rightarrow \frac{a}{ID}=\frac{2a-x}{x}$ $\Rightarrow ID=\frac{ax}{2a-x}$ $\Rightarrow AI=a-\frac{ax}{2a-x}$ $\Rightarrow AI=\frac{a(2a-x)-ax}{2a-x}$ $\Rightarrow AI=\frac{2a^2-2ax}{2a-x}$ $\Rightarrow AI=\frac{2a(a-x)}{2a-x}$ Diện tích tam giác ACI là: $S_{ACI}=\frac{1}{2}\times AI\times AC$ $=\frac{1}{2}\times \frac{2a(a-x)}{2a-x}\times a$ $=\frac{a^2(a-x)}{2a-x}$ Đáp số: $\frac{a^2(a-x)}{2a-x}$