[alogo] Conics enveloping lines

Let c = (AB) be a fixed circle, F be a fixed point, and h = (CDE) be a fixed angle. Let G be a moving point on the circle and H a point dividing segment FG in parts having fixed length-ratio k. Draw angle i = (FHI) equal to (CDE). The envelope of line HI is a conic section, depending on the position of H on FG and the angle (CDE). This generalizes the well known fact for the case where HI is the medial line of FG.

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<-- SCHEME-START -- >
<-- TOOL -- > Select the Circle-tool: (11th-Button).
Click at A, drag ... , release at B, to define circle c, with center at A, passing through B.
<-- TOOL -- > Select the Point-tool: (3rd-Button), (Shortcut: Ctrl+E).
Click at F to define a point there.
<-- TOOL -- > Select the Angle-tool: (7th-Button), (Shortcut: Ctrl+Alt+A).
Click at C, drag ... , release at D. Click again at E, to define angle h = angle(CDE).
<-- TOOL -- > Select the Point-tool: (3rd-Button), (Shortcut: Ctrl+E).
Click next to the contour of circle c, holding down the SHIFT key, to set a point G on the contour of circle c.
<-- TOOL -- > Select the Segment-tool: (5th-Button), (Shortcut: Ctrl+Alt+S).
Click at F, drag ... , release at point G, to define segment e=FG.
<-- TOOL -- > Select the Point-tool: (3rd-Button), (Shortcut: Ctrl+E).
Click next to the contour of segment e, holding down the SHIFT key, to set a point H on the contour of segment e.
<-- TOOL -- > Select the Angle-compasses-tool: (7th-Button/2nd-Menu-item), (Shortcut: Ctrl+Shift+A).
Click at D of angle h, to get its angle. Then click at H and F to transfer the angle so that its first side is HF.
<-- TOOL -- > Select the Line-tool: (6th-Button), (Shortcut: Ctrl+L).
Click at H, drag ... , release at point I, to define line j=HI.
<-- TOOL -- > Select the Envelope-tool: (14th-Button/2nd-menu-item), (Shortcut: Ctrl+4).
Click on point G. Then click on line j, to find the envelope of this line as point G moves on circle c.
<-- SCHEME-END -- >

The above description is automatically produced by selecting the [Describes Schemes] tool (1st button/6th menu-item).

Produced with EucliDraw©