Proposition III.17 — From a given point to draw a straight line touching a given …

Proof

Let

A

be the external point and

B C D

the circle with centre

E

. Join

A E

, and at

E

erect a perpendicular to

A E

(I.11); with

E

as centre and

E A

as radius describe a circle

A F G

, meeting the perpendicular at

F

. Join

F A

, meeting the original circle

B C D

B

. Then

A B

is the desired tangent. Proof:

△ A B E

and

△ A F E

are congruent by SAS (

E A

common,

E B = E F

both equal to the radius of

A F G

∠ A E B = ∠ A E F

by construction); hence

∠ A B E = ∠ A F E

, which is right. So

A B ⊥ E B

, the radius at the point of contact, and by III.18 (next)

A B

is tangent.

Knowledge graph · drag to pan, scroll to zoom, click a node to navigate

Full neighborhood

Depends on (5)

III.1Proposition III.1To find the centre of a given circle.
III.16Proposition III.16The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle,…
I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…

Required by (dependents) (2)

III.34Proposition III.34From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.
III.37Proposition III.37If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut…

Discussion

No replications, contradictions, or comments registered yet for this claim.

Replicate or annotate this claim

Replicate to register a fresh attempt; contradict, extend, or comment otherwise. Authors can post a claim-retraction with the reason taxonomy from RRP-0020.