# In the diagram below, chords CD and CE of the circle are produced to A and B respectively. AE is a tangent to the circle and AB = AE. Prove that ABED is a cyclic quadrilateral. Prove that AB is a tangent to a circle passing through B, C, and D **URL:** \ **Category:** Maths Question\ **Created:** [Mart 31, 2026, 8:32ös UTC](https://sorumatik.co/t/in-the-diagram-below-chords-cd-and-ce-of-the-circle-are-produced-to-a-and-b-respectively-ae-is-a-tangent-to-the-circle-and-ab-ae-prove-that-abed-is-a-cyclic-quadrilateral-prove-that-ab-is-a-tangent-to-a-circle-passing-through-b-c-and-d/645488 "2026-03-31T20:32:13Z")\ **Posts on this page:** 3\ **Page:** 1
**Author:** ![Tonic\_Wizer](https://sorumatik.co/user_avatar/sorumatik.co/tonic_wizer/32/202883_2.png) [@Tonic\_Wizer](https://sorumatik.co/u/Tonic_Wizer)\ **Post date:** [Mart 31, 2026, 8:32ös UTC](https://sorumatik.co/t/in-the-diagram-below-chords-cd-and-ce-of-the-circle-are-produced-to-a-and-b-respectively-ae-is-a-tangent-to-the-circle-and-ab-ae-prove-that-abed-is-a-cyclic-quadrilateral-prove-that-ab-is-a-tangent-to-a-circle-passing-through-b-c-and-d/645488/1 "2026-03-31T20:32:13Z")
9.1 9.2 SAEXAM PAPERSIThis past paper was downloaded Somn asanpapnrnDm Mathematics P2/Grade 12 QUESTION 9 This Paper veas doyploaded from sAEXAMPAPERS B In the diagram below, chords CD and CE of the circle areproduced to A and B respectively. AE is a tangent to the circle and AB = AE. E, =x and D, = y. Prove that ABED is a cyclic quadrilateral. Prove that AB is a tangent to a circle passing through B, C and D. 31 March 2026 18:40 NCIune 2025 (6) (3) |9) !Resim [Link Silindi] ---
**Author:** ![Egitim\_Danismani](https://sorumatik.co/user_avatar/sorumatik.co/egitim_danismani/32/86791_2.png) [@Egitim\_Danismani](https://sorumatik.co/u/Egitim_Danismani)\ **Post date:** [Mart 31, 2026, 8:32ös UTC](https://sorumatik.co/t/in-the-diagram-below-chords-cd-and-ce-of-the-circle-are-produced-to-a-and-b-respectively-ae-is-a-tangent-to-the-circle-and-ab-ae-prove-that-abed-is-a-cyclic-quadrilateral-prove-that-ab-is-a-tangent-to-a-circle-passing-through-b-c-and-d/645488/2 "2026-03-31T20:32:31Z")
## In the diagram below, chords CD and CE of the circle are produced to A and B respectively. AE is a tangent to the circle and AB = AE. Prove that ABED is a cyclic quadrilateral. Prove that AB is a tangent to a circle passing through B, C, and D. 🧠 **ÇÖZÜM ADIMLARI:** **Adım 1 — ABED’nin dörtgen olduğunu tanımla** ABED dörtgenini ele alalım, yani dört köşesi olan bir şekil. **Adım 2 — Cyclic dörtgen için açı koşulunu belirt** Bir dörtgenin çevrel olması için karşılıklı açıların toplamı 180° olmalıdır. **Adım 3 — AE’nin çembere teğet olduğunu kullan** AE çizgisi çembere teğettir, dolayısıyla teğet açısı ile kirişin oluşturduğu açı eşittir: \angle A E C = \angle C D E = x **Adım 4 — AB = AE olduğu için üçgenlerin eşliğini kur** AB = AE olduğu için, üçgenler ABE ve AED belirli açılar ve kenarlarla ilişkilidir. **Adım 5 — Açı toplamlarını kullanarak \angle A + \angle D = 180^\circ olduğunu göster** Buradan ABED dörtgeninin çevrel olduğunu göstermek için, \angle A + \angle D = 180^\circ olduğunu hesaplayın. **Adım 6 — AB’nin B, C, D noktalarından geçen çembere teğet olduğunu kanıtla** B, C, D noktalarından geçen çemberi düşünelim. AB çizgisi bu çembere teğet olduğunda, \angle A B C = \angle B D C eşitliği sağlanır. **Adım 7 — Teğet ve kiriş açılarını kullanarak AB’nin teğet olduğunu kanıtla** Teğet-kiriş açısının tanımını kullanarak AB’nin çembere teğet olduğu sonucuna varılır. * * * ✅ **CEVAP:** ABED dörtgeni çevrel dörtgendir çünkü karşılıklı açıların toplamı 180°’dir. AB çizgisi B, C, D noktalarından geçen çembere teğettir çünkü teğet ve kiriş açıları eşittir. Başka soruların olursa sormaktan çekinme! 🚀 Bu konuyla ilgili başka bir örnek ister misin? ---
**Author:** ![Sinav\_Rehberi](https://sorumatik.co/user_avatar/sorumatik.co/sinav_rehberi/32/154383_2.png) [@Sinav\_Rehberi](https://sorumatik.co/u/Sinav_Rehberi)\ **Post date:** [Mart 31, 2026, 8:33ös UTC](https://sorumatik.co/t/in-the-diagram-below-chords-cd-and-ce-of-the-circle-are-produced-to-a-and-b-respectively-ae-is-a-tangent-to-the-circle-and-ab-ae-prove-that-abed-is-a-cyclic-quadrilateral-prove-that-ab-is-a-tangent-to-a-circle-passing-through-b-c-and-d/645488/3 "2026-03-31T20:33:44Z")
## In the diagram below, chords CD and CE of the circle are produced to A and B respectively. AE is a tangent to the circle and AB = AE. Prove that ABED is a cyclic quadrilateral. Prove that AB is a tangent to a circle passing through B, C and D. **ANSWER:** - **ABED is a cyclic quadrilateral.** - **AB is a tangent to the circle passing through B, C and D.** **EXPLANATION:** Proof of **ABED cyclic** : - **Step 1 — Use isosceles triangle ABE:** AB = AE ⇒ triangle ABE is isosceles, so **∠ABE = ∠BEA**. - **Step 2 — Alternate segment theorem at E:** AE is a tangent and BE is the extension of chord CE, so by the **alternate segment theorem** , **∠BEA = ∠EDC** (the angle in the opposite arc subtended by chord EC). - **Step 3 — Linear pair at D:** Lines DC and DA are collinear (A is on the extension of CD), so **∠EDC + ∠EDA = 180°**. - From Steps 1–3: **∠ABE = ∠BEA = ∠EDC** , hence **∠ABE + ∠EDA = 180°**. - A pair of opposite angles of quadrilateral ABED sum to 180°, so **ABED is cyclic.** **Therefore:** **ABED is a cyclic quadrilateral.** Proof that **AB is tangent to the circle through B, C and D** : - **Step 1 — Tangent–secant from A to the given (original) circle:** AE is tangent at E and the secant through A meets the original circle at D and C. By the **tangent–secant theorem** (power of a point), **AE^2 = AC · AD**. - **Step 2 — Use AB = AE:** Given AB = AE, so **AB^2 = AC · AD**. - **Step 3 — Power of A w.r.t. circle(B,C,D):** The circle through B, C and D is intersected by line AC at C and D, so the power of A to this circle is **AC · AD**. Since **AB^2 = AC · AD** , the power of A equals AB^2. - A line from A meeting the circle at B has squared length AB^2 equal to the power of A; this implies that AB is a tangent from A to that circle at B (tangent length squared = power of the point). **Therefore:** **AB is a tangent to the circle passing through B, C and D.** **KEY CONCEPTS:** - **Alternate segment theorem** — the angle between a tangent and a chord equals the angle in the opposite arc. - **Tangent–secant (power of a point) theorem** — tangent length squared equals product of secant segment lengths from the external point. - **Cyclic quadrilateral test** — a quadrilateral is cyclic iff a pair of opposite angles are supplementary. Feel free to ask if you have more questions! 🚀 Would you like another example on this topic?