Posts in category '04-right-and-345'

GC-04.01 · H. E. Huntley’s Golden Ratio in the 3-4-5 Right Triangle (1970)

In a \(3\text{-}4\text{-}5\) right triangle ABC (\(|BC|=3, |AC|=4, |AB|=5\)), let the internal bisector of \(\angle ABC\) meet leg AC at O. Circle cOC centered at O through C intersects line BO at P and Q such that \(|PQ|/|BP| = |BQ|/|PQ| = \phi\).

GC-04.02 · Johannes Kepler’s Geometric-Progression Right Triangle (1597)

Divide hypotenuse AB at H in the golden ratio (\(|AH|/|HB| = \phi\)), and erect perpendicular HC to semicircle cOB on diameter AB. Right triangle ABC has side lengths in geometric progression \(1 : \sqrt{\phi} : \phi\) and side quadrances in progression \(1 : \phi : \phi^2\).

GC-04.03 · Jo Niemeyer’s Three Equal Leaning Segments

Place segment A1B1 perpendicular to line L at A1, lean equal segment B2A2 from midpoint B2 of A1B1 onto L, and lean a third equal segment B3A3 from midpoint B3 of B2A2 onto L. Then A2 divides A1A3 in the golden ratio: \(|A1A2|/|A2A3| = \phi\).

GC-04.04 · Bùi Quang Tuân’s Van Obel Division in a Right Isosceles Triangle

In right isosceles triangle ABC with hypotenuse midpoint M and midpoint O of altitude AM, circle cOM cuts OC and OB at D and F. Cevians BD and CF concur on OM at E, dividing OM in the golden ratio: \(|OE|/|EM| = \phi\).