The original problem gave rise khổng lồ a few modifications; và each of them has been solved in many, many ways. I"ll be adding solutions và perhaps problems related khổng lồ the original one. Any assistance is welcomed.

Bạn đang xem: Pareto principle   Original problem 60-70 variant Short segments Long segments Isosceles partition     Another partition 40-50 Variant 30-40 variant 20-30 variant 10-20 variant
Original problem

Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 50°. Point E is on side AB such that ∠BCE = 60°. Find the measure of ∠CED.

Solutions

60-70 Variant

Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 60°. Point E is on side AB such that ∠BCE = 70°. Find the measure of ∠CED.

Solutions

Short segments equality

ABC is an isosceles triangle with vertex angle ∠BAC = 20° & AB = AC. Draw ∠BCD = 60°; D lying on AB. Draw an arc with B as center và radius equal to lớn BC. Let this arc cut AC at point E và AB at the point F. Prove that CE = DF.

Solutions

Long segments equality

ABC is an isosceles triangle with vertex angle ∠BAC = 20° and AB = AC. Point E is on AB such that AE = BC. Find the measure of ∠AEC.

Solutions

Solutions

Partition that starts with a base angle bisector

Solutions

40-50 Variant

Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 40°. Point E is on side AB such that ∠BCE = 50°. Find the measure of ∠CED.

Solutions

30-40 Variant

Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 30°. Point E is on side AB such that ∠BCE = 40°. Find the measure of ∠CED.

Solutions

20-30 Variant

Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 20°. Point E is on side AB such that ∠BCE = 30°. Find the measure of ∠CED.

Solution

10-20 Variant

Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 10°. Point E is on side AB such that ∠BCE = 20°. Find the measure of ∠CED.

Xem thêm: Các Chương Toán 12 : Tổng Hợp Kiến Thức, Bài Tập & Giải Bài Tập Toán 12

Solutions

### References

T. Rike, An Intriguing Geometry Problem, Berkeley Math Circle, May 5, 2002.H. S. M. Coxeter và S. L. Greitzer, Geometry Revisited, MAA, 1967.R. Honsberger, Four Minor Gems from Geometry, Mathematical Gems II, MAA, 1976.R. Honsberger, Three Solutions lớn a Variation on an Old Chestnut, Mathematical Chestnuts from Around the World, MAA, 2001.C. Knop, Nine Solutions to lớn One Problem, Kvant, 1993, no 6.R. Leikin, Dividable Triangles - What Are They?, Mathematics Teacher, May 2001, pp. 392-398.V. V. Prasolov, Essays on Numbers và Figures, MAA, 2000. |Contact||Front page||Contents|