Lesson: M9 ========== The Law of Sines ---------------- .Definition. ************* sin asciimath:[alpha=frac{text{secant}}{text{diameter}}]. ************* For this definition to make sense, we need the .Peripheral Angle Theorem. **************** Two peripheral angles subtending the same secant of a circle are equal or supplementary and therefore have the same sine. **************** 'Proof.' Let asciimath:[A] be the center of a circle. First we consider the case where one side is a diameter. Let asciimath:[DB] be a diameter. Then for a point asciimath:[C] on the circle, asciimath:[/_BAC] is an exterior angle of the isoceles △asciimath:[DAC]. [frame="none",cols="^",valign="middle",grid="all"] |=================== |pass:[] |*Click image to view in KSEG.* | |=================== By the Exterior Angle Theorem of Euclid, we have [frame="none",cols="^",valign="middle",grid="none"] |=================== |asciimath:[/_BAC=/_ADC+/_DCA=alpha+alpha=2 alpha]. |=================== So the peripheral angle is half the central angle to the same secant asciimath:[CB]. Recall the Law of Sines from trigonometry: .Law of Sines. **************** [frame="none",cols="^",valign="middle",grid="none"] |=================== |asciimath:[frac{sin\ alpha}{a}=frac{sin\ beta}{b}=frac{sin\ gamma}{c}]. |=================== **************** The Cross Ratio --------------- .Definition. ************* Given four collinear points asciimath:[ABCD] (in any order), the 'cross ratio' asciimath:[CR(ABCD)] is [frame="none",cols=">,^,>",valign="middle",grid="none"] |=================== ||asciimath:[frac{A-B}{A-D}\ frac{C-D}{C-B}\ =\ frac{sin/_BPA}{sin/_DPA}\ frac{sin/_DPC}{sin/_BPC}]|asciimath:[(**)], |=================== where asciimath:[P] is any point not on the line. ************* [frame="none",cols="^",valign="middle",grid="all"] |=================== |pass:[] |*Click image to view in KSEG.* | |=================== 'Proof of asciimath:[(**)].' Apply the Law of Sines to △asciimath:[ABP] then [frame="none",cols="^",valign="middle",grid="none"] |=================== |asciimath:[frac{A-B}{sin/_BPA}=frac{a}{sin/_B}=frac{b}{sin/_A}], |=================== where asciimath:[A-B] is the signed length asciimath:[|A-B|] and asciimath:[sin/_B=sin/_PBA=sin/_CBP] since supplementary angles have the same sine. Now let's calculate [frame="none",cols="^",valign="middle",grid="none"] |=================== |asciimath:[frac{A-B}{A-D}\ frac{C-D}{C-B}\ =\ frac{A-B}{sin/_BPA}\ frac{C-D}{sin/_DPC}\ =\ frac{P-A}{sin/_B}\ frac{P-C}{sin/_D}]. |=================== Similarly, [frame="none",cols="^",valign="middle",grid="none"] |=================== |asciimath:[frac{sin/_BPA}{sin/_DPA}\ frac{sin/_DPC}{sin/_BPC}\ =\ frac{A-D}{sin/_DPA}\ frac{C-B}{sin/_BPC}\ =\ frac{P-A}{sin/_D}\ frac{P-C}{sin/_B}]. |=================== So asciimath:[frac{A-B}{A-D}\ frac{C-D}{C-B}\ =\ frac{sin/_BPA}{sin/_DPA}\ frac{sin/_DPC}{sin/_BPC}\ \ square]. Perspective Rulers ------------------ .Definition. ************* A 'ruler' is a line together with a copy of the real numbers on it. ************* A 'perspective ruler' on a line with a finite vanishing point is obtained with the following procedure: given asciimath:[l] with vanishing point asciimath:[V], - Choose a point asciimath:[P] not on asciimath:[l] on the extension of asciimath:[l] past asciimath:[V]. - Choose a line asciimath:[s\ ||\ (PV)]. - On asciimath:[s] choose a Euclidean ruler. - Transfer this scale through the lens asciimath:[P] to asciimath:[l]. [frame="none",cols="^",valign="middle",grid="all"] |=================== |pass:[] |*Click image to view in KSEG.* | |===================