On a property of a line intersectiing two tangents of a cercle

[OUADJAOUT]

Let (C) a circle . A is a point outside (C). Let (D1), (D2) two tangents issued from A. (D1) is tangent at K and (D2) is tangent at L respectively. Let (D) a straight line passing through A, between (D1) and (D2). The meeting points with (C) are M and N respectively. Let (LP) the parralel at (D) issued from L, where P is the intersection with(C).
Prouve that (PK) intersect [MN] at his middle.

M. Ouadjaout
ENP,Dépt. SF, Lab. LRE
Algeria

[Peter Scales]

Let KQ // LP meet C at Q

Then let LQ meet PK at J

The common perpendicular bisector of LP and KQ ia a diameter of C

>From symmetry J lies on this diameter.

But triangles ALK, JLP and JKQ are similar

: J lies on AMND

: J is mid-point of MN. QED.

Regards, Peter Scales.

[aliannahfth]

The information was very useful.

[OUADJAOUT]

Thank you for this response.

I have not understood this passage of your prouf :

"But triangles ALK, JLP and JKQ are similar; J lies on AMND"

What means the point D ?
Mohamed

[OUADJAOUT]

Please, can you indicate me one situation for where this result is usuful.

Thank you.

Mohamed

[Peter Scales]

D is defined in your original problem statement as an arbitrary point defining a line AD which lies between the tangents AKD1 and ALD2.

Regards, Peter Scales.

[OUADJAOUT]

Thank you for your message.

I note by (D) a line in my original text, not a point.

Best regards.