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[Minecraft]

The ABC triangle inscribed the circle center . I is the mid point of BC . M is a random point on IC ( M differ from C and D ) . D is the intersection of the circle center O and AM . the intersection of the tangent to the circle of a triangle AMI in M with BD , BC in P and Q.
1 ) prove that : DM . IA = MP .IB
2 ) calculating : MP/MQ = ?

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The ABC triangle inscribed the circle center . I is the mid point of BC . M is a random point on IC ( M differ from C and D ) . D is the intersection of the circle center O and AM . the intersection of the tangent to the circle of a triangle AMI in M with BD , BC in P and Q.
1 ) prove that : DM . IA = MP .IB
2 ) calculating : MP/MQ = ?

1)
We have $\widehat{AMP} =\widehat{AMI} +\widehat{IMP}$
$=\widehat{AMI} +\widehat{IAM} =\widehat{AIB}$(1)
$\Leftrightarrow\widehat{PMD} =\widehat{AIC}$ (2)
have $\widehat{MDP} =\widehat{ICA}$ (3)
from (2, 3) then $\triangle PMD\sim\triangle AIC$(angle-angle)
$\Rightarrow\frac{MD}{IC} =\frac{MP}{IA}$
$\Rightarrow MD .IA =MP .IC =MP .IB$ (4)
2)
(1)$\Rightarrow\widehat{DMQ} =\widehat{BIA}$ (5)
have $\widehat{MDQ} =\widehat{IBA}$ (6)
from (5, 6) then $\triangle MDQ\sim\triangle IBA$ (angle-angle)
$\Rightarrow\frac{MD}{IB} =\frac{MQ}{IA}$
$\Rightarrow MD .IA =MQ .IB$ (7)
from(4, 7) then $MP .IB =MQ .IB$
$\Leftrightarrow MP =MQ$