Tangents to a circle from a point P

Learn how to draw the tangent lines to a given circumference passing through an external given point P.

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[gilbertmiya]

Thanks. Provides many insights to circle properties

[anamfatima]

Best youtube channel for geometric maths
Love from India❤

[danthewalkingmanen-dorsetg]

Your work has helped prefect my crop circle making game

[Antoniosanchez-nhmg]

Te quiero arthur me salvas las clases de dibujo tecnico

[architsharma]

I am the first one for once!
I am sure this will be another amazing vid!

[neuraaquaria]

Here's the proof (I'll let T1 = T here, look at 2:49 for diagram):

1) First draw auxiliary line MT

2) MT = MP. They are both radii to the same circle OTP centered at M.
∴ ∠MTP = ∠MPT by converse of isosceles triangle theorem

3) ∠MTP + ∠MPT = ∠TMO by external angles theorem

∴ 2 x ∠MPT = ∠TMO

4) OM = OT. They are both radii to the same circle OTP centered at M.
∴ ∠MOT = ∠MTO by converse of isosceles triangle theorem

5) ∠MOT + ∠MTO + ∠TMO = 180° by triangle angle sum.
∠MOT + ∠MOT + ∠TMO = 180° by substituting step (4)
2 x ∠MOT + ∠TMO = 180°

2 x ∠MOT + 2 x ∠MPT = 180°
∠MOT + ∠MPT = 90° divide both sides by 90
∴∠ OTP = 90° by triangle angle sum

6) ∴ OT ⊥ TP and given that OT is a radius of the circle, TP is tangent to the circle because tangent lines are perpendicular to radius.

[iuliuscaesar]

Hello, I have seen many videos of tangency but I have a problem, I don't want to memorize the steps for each case of tangency, I want to know why by making several strokes the points of tangency are obtained in the drawing. Where should I start to understand the "why" do the centers join, why does it become bisector, why do we have to add radii or subtract? I don't want to learn just the steps to solve tangency problems. Sorry if it's a lot of text, I hope I explained it well.