Advanced Euclidean Geometry

Paul Yiu

Department of Mathematics Florida Atlantic University

Summer 2016 √ 5+1 The Golden Ratio ϕ = 2 Odom1

ABC is an equilateral triangle, with midpoints D, E of the sides AB and AC. The segment DE is extended to intersect the circumcircle at F . E divides DF in the golden ratio.

A

D E F

B C

Yiu: Advanced Euclidean Geometry 2016

1AMER.MATH.MONTHLY, Elementary Problem 3007, volume 90 (1983) 482.

1 M. Bataille2

Given an equilateral triangle ABC, erect a square BCDE externally on the side BC. Construct the circle, center C, passing through E, to intersect the line AB at F . Then, B divides AF in the golden ratio.

D

C

E

A B F

Yiu: Advanced Euclidean Geometry 2016

2Forum Geom., 11 (2011) 55.

2 Niemeyer3

Three equal segments A1B1, A2B2, A3B3 are positioned in such a way that the endpoints B2, B3 are the midpoints of A1B1, A2B2 respectively, while the endpoints A1, A2, A3 are on a line perpendicular to A1B1.

B1

B2

B3

A1 A2 A3

In this arrangement, A2 divides A1A3 in the golden ratio, namely, √ A A 5+1 1 3 = . A1A2 2 Yiu: Advanced Euclidean Geometry 2016 3Forum Geom., 11 (2011) 53

3 Dao4

Consider an equilateral triangle ABC with its sides AC and AB divided into five equal parts by points Ek, Fk, k =1, 2, 3, 4, so that AEk = AFk = k 5 · BC. If the circle (AE4F4) intersects BC at G and H, then G divides HB in the golden ratio.

A

F4 E4

B G H C

Yiu: Advanced Euclidean Geometry 2016

4Forum Geom., 16 (2016) 269Ð272.

4 Tran5

Given a right isosceles triangle ABC and its circumcircle, inscribed a square DEFG with a side FG along the hypotenuse AB. If the side DE is ex- tended to intersect the circumcircle at P , then E divides DP in the golden ratio.

C

D E P

B G F A

Yiu: Advanced Euclidean Geometry 2016

5Forum Geom., 15 (2015) 91Ð92.

5 Dao6

The equal sides AC and AB of a right isosceles triangle ABC are divided k into five equal parts, at Ek, Fk, k =1, 2, 3, 4, so that AEk = AFk = 5 · AB. The circle (AE3F3) intersects BC at F and G. The point G divides HC in the golden ratio.

A

F3 E3

B H G C

Yiu: Advanced Euclidean Geometry 2016

6Forum Geom., 16 (2016) 269Ð272.

6 Equilateral triangle, square, and regular pentagon

An equilateral triangle ABC, a square ABCC, and a regular pentagon ABCDE are constructed on the same side of AB. The side BC of the square divides the segment CC in the golden ratio. D

D C E C C P

A B

Yiu: Advanced Euclidean Geometry 2016

7 A variation

D

D C E C

P

C A B

Yiu: Advanced Euclidean Geometry 2016

8 Proof

D

D C E C Q Q

P

C A B

CC QC EC EC = = = = . CP QQ DC AB golden ratio

Yiu: Advanced Euclidean Geometry 2016

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