Triangle Centers¶

Special points and circles associated with triangles.

API Reference¶

geolet.primitives.centers.api.Centroid ¶

Centroid(p1: PointT, p2: PointT, p3: PointT, *, label: str = '', label_dir: str = 'NE', color: str = 'black') -> IntersectionPoint

Create centroid of a triangle (intersection of medians).

Pure composition: intersects two median segments (with inline midpoints).

Parameters:

Name	Type	Description	Default
`p1`	`PointT`	First vertex of triangle.	required
`p2`	`PointT`	Second vertex of triangle.	required
`p3`	`PointT`	Third vertex of triangle.	required
`label`	`str`	Display name for the centroid (e.g. 'G'). Omit for hidden.	`''`
`label_dir`	`str`	Direction to place label relative to point.	`'NE'`
`color`	`str`	Asymptote color name.	`'black'`

Returns:

Type	Description
`IntersectionPoint`	The centroid point.

geolet.primitives.centers.api.Orthocenter ¶

Orthocenter(p1: PointT, p2: PointT, p3: PointT, *, label: str = '', label_dir: str = 'NE', color: str = 'black') -> IntersectionPoint

Create orthocenter of a triangle (intersection of altitudes).

Pure composition: intersects two altitude lines (perpendiculars from vertices to opposite sides).

Parameters:

Name	Type	Description	Default
`p1`	`PointT`	First vertex of triangle.	required
`p2`	`PointT`	Second vertex of triangle.	required
`p3`	`PointT`	Third vertex of triangle.	required
`label`	`str`	Display name for the orthocenter (e.g. 'H'). Omit for hidden.	`''`
`label_dir`	`str`	Direction to place label relative to point.	`'NE'`
`color`	`str`	Asymptote color name.	`'black'`

Returns:

Type	Description
`IntersectionPoint`	The orthocenter point.

geolet.primitives.centers.api.Circumcenter ¶

Circumcenter(p1: PointT, p2: PointT, p3: PointT, *, label: str = '', label_dir: str = 'NE', color: str = 'black') -> IntersectionPoint

Create circumcenter of a triangle (intersection of perpendicular bisectors).

Pure composition: intersects two perpendicular bisector lines. The circumcenter is equidistant from all three vertices.

Parameters:

Name	Type	Description	Default
`p1`	`PointT`	First vertex of triangle.	required
`p2`	`PointT`	Second vertex of triangle.	required
`p3`	`PointT`	Third vertex of triangle.	required
`label`	`str`	Display name for the circumcenter (e.g. 'O'). Omit for hidden.	`''`
`label_dir`	`str`	Direction to place label relative to point.	`'NE'`
`color`	`str`	Asymptote color name.	`'black'`

Returns:

Type	Description
`IntersectionPoint`	The circumcenter point.

geolet.primitives.centers.api.Incenter ¶

Incenter(p1: PointT, p2: PointT, p3: PointT, *, label: str = '', label_dir: str = 'NE', color: str = 'black') -> IntersectionPoint

Create incenter of a triangle (intersection of angle bisectors).

Pure composition: intersects two angle bisector lines.

Parameters:

Name	Type	Description	Default
`p1`	`PointT`	First vertex of triangle.	required
`p2`	`PointT`	Second vertex of triangle.	required
`p3`	`PointT`	Third vertex of triangle.	required
`label`	`str`	Display name for the incenter (e.g. 'I'). Omit for hidden.	`''`
`label_dir`	`str`	Direction to place label relative to point.	`'NE'`
`color`	`str`	Asymptote color name.	`'black'`

Returns:

Type	Description
`IntersectionPoint`	The incenter point.

geolet.primitives.centers.api.Incircle ¶

Incircle(p1: PointT, p2: PointT, p3: PointT, *, color: str = 'black', style: str = 'solid', width: float = 1.0) -> CircleT

Create incircle of a triangle (inscribed circle).

Pure composition: Incenter -> Foot -> CircleThroughPoint. The incircle has its center at the incenter and passes through the foot of the perpendicular from incenter to any side.

Parameters:

Name	Type	Description	Default
`p1`	`PointT`	First vertex of triangle.	required
`p2`	`PointT`	Second vertex of triangle.	required
`p3`	`PointT`	Third vertex of triangle.	required
`color`	`str`	Asymptote color name.	`'black'`
`style`	`str`	Line style. One of: solid, dashed, dotted.	`'solid'`
`width`	`float`	Line width in Asymptote units.	`1.0`

Returns:

Type	Description
`CircleT`	The incircle.

geolet.primitives.centers.api.Median ¶

Median(vertex: PointT, p1: PointT, p2: PointT, *, color: str = 'black', style: str = 'solid', width: float = 1.0) -> SegmentT

Create a median of a triangle (segment from vertex to midpoint of opposite side).

Pure composition: Segment(vertex, Midpoint(p1, p2)).

Parameters:

Name	Type	Description	Default
`vertex`	`PointT`	The vertex the median starts from.	required
`p1`	`PointT`	First endpoint of the opposite side.	required
`p2`	`PointT`	Second endpoint of the opposite side.	required
`color`	`str`	Asymptote color name.	`'black'`
`style`	`str`	Line style. One of: solid, dashed, dotted.	`'solid'`
`width`	`float`	Line width in Asymptote units.	`1.0`

Returns:

Type	Description
`SegmentT`	The median segment.

geolet.primitives.centers.api.Foot ¶

Foot(point: PointT, to: SegmentT | LineT, *, label: str = '', label_dir: str = 'NE', color: str = 'black') -> IntersectionPoint

Create foot of perpendicular from a point to a line or segment.

The foot is where the perpendicular from point meets the line. This is pure composition: Intersection(Line(p1,p2), PerpendicularLine(point, to)).

Parameters:

Name	Type	Description	Default
`point`	`PointT`	The point from which the perpendicular is drawn.	required
`to`	`SegmentT \| LineT`	The line or segment to which the perpendicular is drawn.	required
`label`	`str`	Display name for the foot (e.g. 'H'). Omit for hidden.	`''`
`label_dir`	`str`	Direction to place label relative to point.	`'NE'`
`color`	`str`	Asymptote color name.	`'black'`

Returns:

Type	Description
`IntersectionPoint`	The foot intersection point.

Examples¶

Triangle Centers via Triangle Methods¶

The easiest way to access triangle centers is through Triangle methods:

from geolet import Point, Triangle, autofigure


@autofigure
def triangle_centers():
    A = Point("A", 0, 0, label_dir="SW")
    B = Point("B", 6, 0, label_dir="SE")
    C = Point("C", 2, 4, label_dir="N")

    T = Triangle(A, B, C)

    T.centroid(label="G", color="blue")
    T.orthocenter(label="H", color="red")
    T.circumcenter(label="O", color="green")
    T.incenter(label="I", color="purple")

Euler Line¶

The orthocenter, centroid, and circumcenter lie on a line:

from geolet import Point, Triangle, Line, autofigure


@autofigure
def euler_line():
    A = Point("A", 0, 0, label_dir="SW")
    B = Point("B", 6, 0, label_dir="SE")
    C = Point("C", 2, 5, label_dir="N")

    T = Triangle(A, B, C)

    H = T.orthocenter(label="H", color="red")
    G = T.centroid(label="G", color="blue")
    O = T.circumcenter(label="O", color="green")

    Line(H, O, color="purple", style="dashed")

Incircle and Circumcircle¶

from geolet import Point, Triangle, autofigure


@autofigure
def triangle_circles():
    A = Point("A", 0, 0, label_dir="SW")
    B = Point("B", 5, 0, label_dir="SE")
    C = Point("C", 2, 4, label_dir="N")

    T = Triangle(A, B, C)

    T.incircle(color="blue")
    T.incenter(label="I", color="blue")

    T.circumcircle(color="red")
    T.circumcenter(label="O", color="red")

Altitude and Foot¶

from geolet import Point, Segment, Foot, autofigure


@autofigure
def altitude():
    A = Point("A", 1, 3, label_dir="N")
    B = Point("B", 0, 0, label_dir="SW")
    C = Point("C", 4, 0, label_dir="SE")

    Segment(A, B)
    bc = Segment(B, C)
    Segment(C, A)

    # Foot of altitude from A to BC
    H = Foot(A, bc, label="H", label_dir="S")
    Segment(A, H, style="dashed", color="blue")