*Notes to a video lecture on http://www.unizor.com*

__Geometry+ 03__

This lecture is dedicated to problems of triangle construction by its certain elements.

The tools of construction are a ruler to draw straight lines and a compass to draw circles.

We will use the following naming rules.

Vertices of triangle will be call by upper case Latin letters

*,*

**A***and*

**B***.*

**C**Sides will be called by lower case Latin letters corresponding to opposite vertices: side

*is opposite to vertex*

**a***etc.*

**A**Angles will be called by lower case Greek letters corresponding to names of their vertices: angle

*is at vertex*

**α***etc.*

**A**Medians are named

*with a subscript of a side onto which they fall: median*

**m***is from vertex*

**m**_{a}*to side*

**A***etc.*

**a**Altitudes are named

*with similar subscripts, like*

**h***etc.*

**h**_{a}Angle bisectors are named

*etc.*

**l**_{a}Radius of a circumscribed circle of a triangle is named

*.*

**R**Radius of an inscribed circle is named

*.*

**r***Problem A*

Construct a triangle by its three altitudes

*,*

**h**_{a}*and*

**h**_{b}*.*

**h**_{c}*Analysis A*

As we know, a product of a side by an altitude falling on it is a double area of a triangle.

Therefore,

**a·h**_{a}= b·h_{b}= c·h_{c}Hence,

**b = a·h**_{a}/h_{b}

**c = a·h**_{a}/h_{c}Let's construct a triangle similar to Δ

*defined by sides*

**ABC***,*

**a***and*

**b***by*

**c****choosing any segment**

*and defining two other segments*

**x***and*

**y***using the equations similar to above.*

**z**

**y ≝ x·h**_{a}/h_{b}

**z ≝ x·h**_{a}/h_{c}Segments

*and*

**y***can be easily constructed from these definitions, knowing*

**z***(arbitrarily chosen) and given altitudes.*

**x**Let

*be a scaling factor between arbitrarily chosen segment*

**x/a=k***and side*

**x***of triangle Δ*

**a***.*

**ABC**From this follow these relationships:

**x = a·k**

**y = x·h**_{a}/h_{b}= a·k·h_{a}/h_{b}= b·k

**z = x·h**_{a}/h_{c}= a·k·h_{a}/h_{c}= c·kThe scaling factor

*is the same for*

**k***,*

**x/a=k***and*

**y/b=k***.*

**z/c=k**Therefore, triangle Δ

*constructed from three segments*

**XYZ***,*

**x***and*

**y***is*

**z****similar**to triangle Δ

*with segments*

**ABC***,*

**a***and*

**b***we have to construct.*

**c**From similarity of triangles follows the congruence of corresponding angles

*∠*

*∠*

**BAC =***∠*

**α =***∠*

**YXZ**

*∠*

**CBA =***∠*

**β =***∠*

**ZYX**

*∠*

**ACB =***∠*

**γ =**

**XZY**Therefore, our analysis shows that by constructing Δ

*we get all angles of Δ*

**XYZ***.*

**ABC**This is the end of analysis, as the construction of triangle Δ

*, knowing its three angles and altitudes, is straight forward.*

**ABC***Solution A*

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