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Chords, secants and tangents of a circle

الأسبوع 8 · تطبيقات التشابه في الدائرة: الوتران المتقاطعان والقاطعان والمماس وعكس ذلك · 3 دروس
3

The converses: four points on one circle, a cyclic quadrilateral and a line that touches a circle

Mathematics Sec 1 Term 1, Unit 2, Lesson 4
GoalUse the equal products to prove that four points lie on a circle or that a line is a tangent.
New words
cyclic quadrilateral · a four-sided figure whose four vertices lie on one circleconverse · the statement turned round: if the product rule is true, the points are concyclic
Explanation

The theorems also work backwards. Converse of the chord theorem: if the lines AB and CD meet at E and EA × EB = EC × ED (with E inside both segments, or outside both), then A, B, C, D lie on one circle. Why: the equal products give EA/ED = EC/EB; the angle at E is common (or vertical), so △EAC ~ △EDB by SAS, which makes ∠EAC = ∠EDB. Two equal angles looking at the same segment from the same side show that the four points are concyclic. Converse of the tangent theorem: if (AB)² = AC × AD, with A, C, D on a line, then AB touches the circle through B, C and D. To use a converse, calculate both products and compare them. If they are different, the points are not on one circle.

If you knowthen
EA × EB = EC × EDA, B, C, D lie on one circle
AD × AB = AE × ACD, B, C, E lie on one circle (DBCE is a cyclic quadrilateral)
(AB)² = AC × ADAB touches the circle through B, C, D
Worked examples
D is on AB and E is on AC with AB = 10, AD = 6, AC = 12 and AE = 5. AD × AB = 6 × 10 = 60 and AE × AC = 5 × 12 = 60. The products are equal, so D, B, C, E lie on one circle: DBCE is a cyclic quadrilateral.
Chords meeting at E inside: EA = 3, EB = 8, EC = 4 and ED = 6. 3 × 8 = 24 = 4 × 6, so A, B, C, D are concyclic. If ED were 5, then 4 × 5 = 20 ≠ 24 and the four points would not lie on one circle.
A, C, D are on a line with C between A and D. AB = 6, AC = 4 and AD = 9. AC × AD = 36 = AB². So AB touches the circle through B, C, D at B.
Class activity

Test your own points. Draw two lines meeting at E. Mark A, B on one line and C, D on the other so that EA × EB = EC × ED (for example 2 × 6 and 3 × 4). Then draw a circle through A, B, C: does it pass through D?

Practice
1
D is on AB, E on AC: AD = 3, AB = 16, AE = 4, AC = 12. Are D, B, C, E concyclic?
2
Chords meet inside at E: EA = 3, EB = 8, EC = 5 and ED = 4. Are A, B, C, D concyclic?
3
AB = 10, AC = 5, AD = 20 with A, C, D on a line. Is AB a tangent to the circle through B, C, D?
4
A, C, D are on a line with AC = 3 and AD = 12. AB = 5. Is AB a tangent to the circle through B, C, D?
5
Why must we compare the two products before saying four points are concyclic?