<@&776327020388679680> I need someone help me figure what is involution and what is look like and also i want to know about DDIT and its applications too, i have read markbcc 168 paper but don't understand
#Involution ,DIT ,DDIT
digital trench
finite plover
Which part don't you understand
digital trench
can you give me an example of a point "A" after involution
i don't understand what involution be like
finite plover
Let L be a line
Let O be a fixed point on L
f that send A to A' on the line L such that $OA \cdot OA' = 1$ is an involution
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digital trench
wow ok
finite plover
I know him personally. You can tell me what you don't understand
digital trench
🙏 wow sir
thank you
in the pdf he defines, Let P, A, B, C, D be points on a plane with AB ∩ CD = E,AD ∩ BC = F. Let a conic C tangent to lines AB,CD,AD,BC. Let PX,PY are the tangent line from P to C. Then (PX,PY),(PA,PC),(PB,PD),(PE,PF) are reciprocal pairs of some involution on pencil of lines pass through P.
in the ddit i don't understand the italic part
why can we involution on lines
finite plover
you understand the definition of involution of points on line right?
digital trench
yes
finite plover
In this case, it is involution of lines passing through a fixed point P. Let f is a funtion of those lines.
suppose that there is a fixed line L not passing through P
Let g be a function of points on line L such that
$$g(X) = L \cap f(\overline{PX}) \quad \forall X \in L$$
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finite plover
Then f is an involution of lines if and only if g is an involution of points
To make it easy, we can just say that the projection of involution of lines on a line is an involution of points.
digital trench
ok
okok
Let ABC and DEF be two triangles which share an incircle ω and cir- cumcircle γ. Let L be the tangency point of EF on ω and define K similarly on BC. Select N ≡ AL ∩ γ and M ≡ DK ∩ γ. Show that lines AM, EF, BC, ND are concurrent.
in this problem they said (DA,DM) is a pair
but idk why
finite plover
in this problem they said (DA,DM) is a pair
pair of which involution
finite plover
when you talk about involution, there must exist 3 pairs to uniquely determine an involution
digital trench
but does the tangency at K do anything
finite plover
swapping DAto DM DE to DF and DB to DC
He mentions DDIT on a triangle previously
finite plover
it is
(DB,DC), (DE,DF),(DA,DK)
finite plover
oh so K is the special case?
DDIT triangle is an special case of DDIT quadrilateral
digital trench
ah yes
i understood now
By theorem 3, lines AM,EF,BC are concurrent. Similarly EF,BC,ND are concurrent
and we are done.
what is theorem 3 sir
finite plover
Line $\overline{AA'}$ passes through a fixed point.
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finite plover
when $A' = f(A)$ when $f$ is an involution on conic
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