Question

Mn giải giúp mình c4 vs ạ. Mình cảm ơn nhiều

Answer 1

Gọi O là trung điểm IK \(\Rightarrow OI=OK=\dfrac{1}{2}IK\)

\(\left(\overrightarrow{MI}+\overrightarrow{IA}\right)\left(\overrightarrow{MI}+\overrightarrow{IB}\right)+\left(\overrightarrow{MK}+\overrightarrow{KC}\right)\left(\overrightarrow{MK}+\overrightarrow{KD}\right)=\dfrac{1}{2}Ik^2\)

\(\Leftrightarrow MI^2-IA^2+MK^2-KC^2=\dfrac{1}{2}IK^2\)

\(\Leftrightarrow\left(\overrightarrow{MO}+\overrightarrow{OI}\right)^2+\left(\overrightarrow{MO}+\overrightarrow{OK}\right)^2=IA^2+KC^2+\dfrac{1}{2}IK^2\)

\(\Leftrightarrow2MO^2+2OI^2=IA^2+KC^2+\dfrac{1}{2}IK^2\)

\(\Leftrightarrow2MO^2+\dfrac{1}{2}IK^2=IA^2+KC^2+\dfrac{1}{2}IK^2\)

\(\Leftrightarrow MO^2=\dfrac{1}{2}\left(IA^2+KC^2\right)=\dfrac{1}{8}\left(AB^2+CD^2\right)\)

\(\Leftrightarrow MO=\dfrac{1}{2\sqrt{2}}\sqrt{AB^2+CD^2}\)

Tập hợp M là đường tròn tâm O bán kính \(\dfrac{\sqrt{AB^2+CD^2}}{2\sqrt{2}}\)